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Theorem fpwrelmap 30536
 Description: Define a canonical mapping between functions from 𝐴 into subsets of 𝐵 and the relations with domain 𝐴 and range within 𝐵. Note that the same relation is used in axdc2lem 9874 and marypha2lem1 8898. (Contributed by Thierry Arnoux, 28-Aug-2017.)
Hypotheses
Ref Expression
fpwrelmap.1 𝐴 ∈ V
fpwrelmap.2 𝐵 ∈ V
fpwrelmap.3 𝑀 = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
Assertion
Ref Expression
fpwrelmap 𝑀:(𝒫 𝐵m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝐵,𝑓,𝑥,𝑦
Allowed substitution hints:   𝑀(𝑥,𝑦,𝑓)

Proof of Theorem fpwrelmap
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fpwrelmap.3 . . 3 𝑀 = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
2 fpwrelmap.1 . . . . . 6 𝐴 ∈ V
32a1i 11 . . . . 5 (𝑓 ∈ (𝒫 𝐵m 𝐴) → 𝐴 ∈ V)
4 simpr 488 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦 ∈ (𝑓𝑥))
5 elmapi 8426 . . . . . . . . . . 11 (𝑓 ∈ (𝒫 𝐵m 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
65ffvelrnda 6835 . . . . . . . . . 10 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ 𝒫 𝐵)
76adantr 484 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → (𝑓𝑥) ∈ 𝒫 𝐵)
8 elelpwi 4511 . . . . . . . . 9 ((𝑦 ∈ (𝑓𝑥) ∧ (𝑓𝑥) ∈ 𝒫 𝐵) → 𝑦𝐵)
94, 7, 8syl2anc 587 . . . . . . . 8 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦𝐵)
109ex 416 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
1110alrimiv 1928 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → ∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
12 abss 3989 . . . . . . 7 ({𝑦𝑦 ∈ (𝑓𝑥)} ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
13 fpwrelmap.2 . . . . . . . 8 𝐵 ∈ V
1413ssex 5192 . . . . . . 7 ({𝑦𝑦 ∈ (𝑓𝑥)} ⊆ 𝐵 → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
1512, 14sylbir 238 . . . . . 6 (∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵) → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
1611, 15syl 17 . . . . 5 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
173, 16opabex3d 7658 . . . 4 (𝑓 ∈ (𝒫 𝐵m 𝐴) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ∈ V)
1817adantl 485 . . 3 ((⊤ ∧ 𝑓 ∈ (𝒫 𝐵m 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ∈ V)
192mptex 6970 . . . 4 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ V
2019a1i 11 . . 3 ((⊤ ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ V)
2110imdistanda 575 . . . . . . . . . 10 (𝑓 ∈ (𝒫 𝐵m 𝐴) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → (𝑥𝐴𝑦𝐵)))
2221ssopab2dv 5406 . . . . . . . . 9 (𝑓 ∈ (𝒫 𝐵m 𝐴) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
2322adantr 484 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
24 simpr 488 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
25 df-xp 5528 . . . . . . . . 9 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
2625a1i 11 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
2723, 24, 263sstr4d 3963 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 ⊆ (𝐴 × 𝐵))
28 velpw 4504 . . . . . . 7 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑟 ⊆ (𝐴 × 𝐵))
2927, 28sylibr 237 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
305feqmptd 6715 . . . . . . . 8 (𝑓 ∈ (𝒫 𝐵m 𝐴) → 𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
3130adantr 484 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
32 nfv 1915 . . . . . . . . 9 𝑥 𝑓 ∈ (𝒫 𝐵m 𝐴)
33 nfopab1 5102 . . . . . . . . . 10 𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
3433nfeq2 2972 . . . . . . . . 9 𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
3532, 34nfan 1900 . . . . . . . 8 𝑥(𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
36 df-rab 3115 . . . . . . . . . 10 {𝑦𝐵𝑥𝑟𝑦} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)}
3736a1i 11 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝐵𝑥𝑟𝑦} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)})
38 nfv 1915 . . . . . . . . . . . 12 𝑦 𝑓 ∈ (𝒫 𝐵m 𝐴)
39 nfopab2 5103 . . . . . . . . . . . . 13 𝑦{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
4039nfeq2 2972 . . . . . . . . . . . 12 𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
4138, 40nfan 1900 . . . . . . . . . . 11 𝑦(𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
42 nfv 1915 . . . . . . . . . . 11 𝑦 𝑥𝐴
4341, 42nfan 1900 . . . . . . . . . 10 𝑦((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴)
449adantllr 718 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦𝐵)
45 df-br 5034 . . . . . . . . . . . . . . . . 17 (𝑥𝑟𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟)
46 eleq2 2878 . . . . . . . . . . . . . . . . . 18 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
47 opabidw 5380 . . . . . . . . . . . . . . . . . 18 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥)))
4846, 47syl6bb 290 . . . . . . . . . . . . . . . . 17 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
4945, 48syl5bb 286 . . . . . . . . . . . . . . . 16 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
5049ad2antlr 726 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
51 elfvdm 6684 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑓𝑥) → 𝑥 ∈ dom 𝑓)
5251adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥 ∈ dom 𝑓)
535fdmd 6502 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (𝒫 𝐵m 𝐴) → dom 𝑓 = 𝐴)
5453adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → dom 𝑓 = 𝐴)
5552, 54eleqtrd 2892 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝐴)
5655ex 416 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ (𝒫 𝐵m 𝐴) → (𝑦 ∈ (𝑓𝑥) → 𝑥𝐴))
5756pm4.71rd 566 . . . . . . . . . . . . . . . 16 (𝑓 ∈ (𝒫 𝐵m 𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
5857ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
5950, 58bitr4d 285 . . . . . . . . . . . . . 14 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦𝑦 ∈ (𝑓𝑥)))
6059biimpar 481 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦)
6144, 60jca 515 . . . . . . . . . . . 12 ((((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → (𝑦𝐵𝑥𝑟𝑦))
6261ex 416 . . . . . . . . . . 11 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) → (𝑦𝐵𝑥𝑟𝑦)))
6359biimpd 232 . . . . . . . . . . . 12 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦𝑦 ∈ (𝑓𝑥)))
6463adantld 494 . . . . . . . . . . 11 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → ((𝑦𝐵𝑥𝑟𝑦) → 𝑦 ∈ (𝑓𝑥)))
6562, 64impbid 215 . . . . . . . . . 10 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
6643, 65abbid 2864 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)})
67 abid2 2932 . . . . . . . . . 10 {𝑦𝑦 ∈ (𝑓𝑥)} = (𝑓𝑥)
6867a1i 11 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} = (𝑓𝑥))
6937, 66, 683eqtr2rd 2840 . . . . . . . 8 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑓𝑥) = {𝑦𝐵𝑥𝑟𝑦})
7035, 69mpteq2da 5127 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
7131, 70eqtrd 2833 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
7229, 71jca 515 . . . . 5 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
73 ssrab2 4008 . . . . . . . . . . . 12 {𝑦𝐵𝑥𝑟𝑦} ⊆ 𝐵
7413, 73elpwi2 5216 . . . . . . . . . . 11 {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵
7574a1i 11 . . . . . . . . . 10 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥𝐴) → {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵)
7675fmpttd 6863 . . . . . . . . 9 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
7776adantr 484 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
78 simpr 488 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
7978feq1d 6477 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓:𝐴⟶𝒫 𝐵 ↔ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵))
8077, 79mpbird 260 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓:𝐴⟶𝒫 𝐵)
8113pwex 5249 . . . . . . . 8 𝒫 𝐵 ∈ V
8281, 2elmap 8433 . . . . . . 7 (𝑓 ∈ (𝒫 𝐵m 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵)
8380, 82sylibr 237 . . . . . 6 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓 ∈ (𝒫 𝐵m 𝐴))
84 elpwi 4508 . . . . . . . . . 10 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → 𝑟 ⊆ (𝐴 × 𝐵))
8584adantr 484 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵))
86 xpss 5538 . . . . . . . . 9 (𝐴 × 𝐵) ⊆ (V × V)
8785, 86sstrdi 3928 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (V × V))
88 df-rel 5529 . . . . . . . 8 (Rel 𝑟𝑟 ⊆ (V × V))
8987, 88sylibr 237 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → Rel 𝑟)
90 relopab 5663 . . . . . . . 8 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
9190a1i 11 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
92 id 22 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
93 nfv 1915 . . . . . . . . 9 𝑥 𝑟 ∈ 𝒫 (𝐴 × 𝐵)
94 nfmpt1 5131 . . . . . . . . . 10 𝑥(𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
9594nfeq2 2972 . . . . . . . . 9 𝑥 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
9693, 95nfan 1900 . . . . . . . 8 𝑥(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
97 nfv 1915 . . . . . . . . 9 𝑦 𝑟 ∈ 𝒫 (𝐴 × 𝐵)
9842nfci 2939 . . . . . . . . . . 11 𝑦𝐴
99 nfrab1 3337 . . . . . . . . . . 11 𝑦{𝑦𝐵𝑥𝑟𝑦}
10098, 99nfmpt 5130 . . . . . . . . . 10 𝑦(𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
101100nfeq2 2972 . . . . . . . . 9 𝑦 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
10297, 101nfan 1900 . . . . . . . 8 𝑦(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
103 nfcv 2955 . . . . . . . 8 𝑥𝑟
104 nfcv 2955 . . . . . . . 8 𝑦𝑟
105 brelg 30414 . . . . . . . . . . . . . . . 16 ((𝑟 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
10684, 105sylan 583 . . . . . . . . . . . . . . 15 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
107106adantlr 714 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
108107simpld 498 . . . . . . . . . . . . 13 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥𝐴)
109107simprd 499 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦𝐵)
110 simpr 488 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥𝑟𝑦)
11178fveq1d 6654 . . . . . . . . . . . . . . . . . 18 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓𝑥) = ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥))
11213rabex 5202 . . . . . . . . . . . . . . . . . . 19 {𝑦𝐵𝑥𝑟𝑦} ∈ V
113 eqid 2798 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
114113fvmpt2 6763 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝐴 ∧ {𝑦𝐵𝑥𝑟𝑦} ∈ V) → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥) = {𝑦𝐵𝑥𝑟𝑦})
115112, 114mpan2 690 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥) = {𝑦𝐵𝑥𝑟𝑦})
116111, 115sylan9eq 2853 . . . . . . . . . . . . . . . . 17 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑓𝑥) = {𝑦𝐵𝑥𝑟𝑦})
117116eleq2d 2875 . . . . . . . . . . . . . . . 16 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ 𝑦 ∈ {𝑦𝐵𝑥𝑟𝑦}))
118 rabid 3331 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑦𝐵𝑥𝑟𝑦} ↔ (𝑦𝐵𝑥𝑟𝑦))
119117, 118syl6bb 290 . . . . . . . . . . . . . . 15 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
120108, 119syldan 594 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
121109, 110, 120mpbir2and 712 . . . . . . . . . . . . 13 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦 ∈ (𝑓𝑥))
122108, 121jca 515 . . . . . . . . . . . 12 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦 ∈ (𝑓𝑥)))
123122ex 416 . . . . . . . . . . 11 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝑟𝑦 → (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
124119simplbda 503 . . . . . . . . . . . 12 ((((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦)
125124expl 461 . . . . . . . . . . 11 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦))
126123, 125impbid 215 . . . . . . . . . 10 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
12745, 126bitr3id 288 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
128127, 47bitr4di 292 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
12996, 102, 103, 104, 33, 39, 128eqrelrd2 30421 . . . . . . 7 (((Rel 𝑟 ∧ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
13089, 91, 92, 129syl21anc 836 . . . . . 6 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
13183, 130jca 515 . . . . 5 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
13272, 131impbii 212 . . . 4 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
133132a1i 11 . . 3 (⊤ → ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))))
1341, 18, 20, 133f1od 7385 . 2 (⊤ → 𝑀:(𝒫 𝐵m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵))
135134mptru 1545 1 𝑀:(𝒫 𝐵m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  {cab 2776  {crab 3110  Vcvv 3441   ⊆ wss 3882  𝒫 cpw 4499  ⟨cop 4533   class class class wbr 5033  {copab 5095   ↦ cmpt 5113   × cxp 5520  dom cdm 5522  Rel wrel 5527  ⟶wf 6325  –1-1-onto→wf1o 6328  ‘cfv 6329  (class class class)co 7142   ↑m cmap 8404 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-ov 7145  df-oprab 7146  df-mpo 7147  df-1st 7681  df-2nd 7682  df-map 8406 This theorem is referenced by:  fpwrelmapffs  30537
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