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Theorem f1oweALT 7297
Description: Alternate proof of f1owe 6744, more direct since not using the isomorphism predicate, but requiring ax-un 7094. (Contributed by NM, 4-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
f1oweALT.1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹𝑥)𝑆(𝐹𝑦)}
Assertion
Ref Expression
f1oweALT (𝐹:𝐴1-1-onto𝐵 → (𝑆 We 𝐵𝑅 We 𝐴))
Distinct variable groups:   𝑥,𝑦,𝑆   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem f1oweALT
Dummy variables 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1ofo 6283 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
2 df-fo 6035 . . . . 5 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
3 freq2 5220 . . . . . . 7 (ran 𝐹 = 𝐵 → (𝑆 Fr ran 𝐹𝑆 Fr 𝐵))
43biimprd 238 . . . . . 6 (ran 𝐹 = 𝐵 → (𝑆 Fr 𝐵𝑆 Fr ran 𝐹))
5 df-fn 6032 . . . . . . 7 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6 df-fr 5208 . . . . . . . . . . . . . . . . . . . 20 (𝑆 Fr ran 𝐹 ↔ ∀𝑤((𝑤 ⊆ ran 𝐹𝑤 ≠ ∅) → ∃𝑢𝑤𝑓𝑤 ¬ 𝑓𝑆𝑢))
7 vex 3354 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ V
87funimaex 6114 . . . . . . . . . . . . . . . . . . . . 21 (Fun 𝐹 → (𝐹𝑧) ∈ V)
9 n0 4078 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ≠ ∅ ↔ ∃𝑤 𝑤𝑧)
10 funfvima2 6634 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (𝑤𝑧 → (𝐹𝑤) ∈ (𝐹𝑧)))
11 ne0i 4069 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐹𝑤) ∈ (𝐹𝑧) → (𝐹𝑧) ≠ ∅)
1210, 11syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (𝑤𝑧 → (𝐹𝑧) ≠ ∅))
1312exlimdv 2013 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (∃𝑤 𝑤𝑧 → (𝐹𝑧) ≠ ∅))
149, 13syl5bi 232 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (𝑧 ≠ ∅ → (𝐹𝑧) ≠ ∅))
1514imp 393 . . . . . . . . . . . . . . . . . . . . . . 23 (((Fun 𝐹𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → (𝐹𝑧) ≠ ∅)
16 imassrn 5616 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹𝑧) ⊆ ran 𝐹
1715, 16jctil 509 . . . . . . . . . . . . . . . . . . . . . 22 (((Fun 𝐹𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → ((𝐹𝑧) ⊆ ran 𝐹 ∧ (𝐹𝑧) ≠ ∅))
18 sseq1 3775 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝐹𝑧) → (𝑤 ⊆ ran 𝐹 ↔ (𝐹𝑧) ⊆ ran 𝐹))
19 neeq1 3005 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝐹𝑧) → (𝑤 ≠ ∅ ↔ (𝐹𝑧) ≠ ∅))
2018, 19anbi12d 616 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = (𝐹𝑧) → ((𝑤 ⊆ ran 𝐹𝑤 ≠ ∅) ↔ ((𝐹𝑧) ⊆ ran 𝐹 ∧ (𝐹𝑧) ≠ ∅)))
21 raleq 3287 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝐹𝑧) → (∀𝑓𝑤 ¬ 𝑓𝑆𝑢 ↔ ∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
2221rexeqbi1dv 3296 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = (𝐹𝑧) → (∃𝑢𝑤𝑓𝑤 ¬ 𝑓𝑆𝑢 ↔ ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
2320, 22imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝐹𝑧) → (((𝑤 ⊆ ran 𝐹𝑤 ≠ ∅) → ∃𝑢𝑤𝑓𝑤 ¬ 𝑓𝑆𝑢) ↔ (((𝐹𝑧) ⊆ ran 𝐹 ∧ (𝐹𝑧) ≠ ∅) → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
2423spcgv 3444 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝑧) ∈ V → (∀𝑤((𝑤 ⊆ ran 𝐹𝑤 ≠ ∅) → ∃𝑢𝑤𝑓𝑤 ¬ 𝑓𝑆𝑢) → (((𝐹𝑧) ⊆ ran 𝐹 ∧ (𝐹𝑧) ≠ ∅) → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
2517, 24syl7 74 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑧) ∈ V → (∀𝑤((𝑤 ⊆ ran 𝐹𝑤 ≠ ∅) → ∃𝑢𝑤𝑓𝑤 ¬ 𝑓𝑆𝑢) → (((Fun 𝐹𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
268, 25syl 17 . . . . . . . . . . . . . . . . . . . 20 (Fun 𝐹 → (∀𝑤((𝑤 ⊆ ran 𝐹𝑤 ≠ ∅) → ∃𝑢𝑤𝑓𝑤 ¬ 𝑓𝑆𝑢) → (((Fun 𝐹𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
276, 26syl5bi 232 . . . . . . . . . . . . . . . . . . 19 (Fun 𝐹 → (𝑆 Fr ran 𝐹 → (((Fun 𝐹𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
2827com23 86 . . . . . . . . . . . . . . . . . 18 (Fun 𝐹 → (((Fun 𝐹𝑧 ⊆ dom 𝐹) ∧ 𝑧 ≠ ∅) → (𝑆 Fr ran 𝐹 → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
2928expd 400 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 → ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (𝑧 ≠ ∅ → (𝑆 Fr ran 𝐹 → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))))
3029anabsi5 648 . . . . . . . . . . . . . . . 16 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (𝑧 ≠ ∅ → (𝑆 Fr ran 𝐹 → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢)))
3130impd 396 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → ((𝑧 ≠ ∅ ∧ 𝑆 Fr ran 𝐹) → ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
32 fores 6263 . . . . . . . . . . . . . . . 16 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (𝐹𝑧):𝑧onto→(𝐹𝑧))
33 fvres 6346 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣𝑧 → ((𝐹𝑧)‘𝑣) = (𝐹𝑣))
34 fvres 6346 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤𝑧 → ((𝐹𝑧)‘𝑤) = (𝐹𝑤))
3533, 34breqan12rd 4803 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤𝑧𝑣𝑧) → (((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤) ↔ (𝐹𝑣)𝑆(𝐹𝑤)))
36 vex 3354 . . . . . . . . . . . . . . . . . . . . . 22 𝑣 ∈ V
37 vex 3354 . . . . . . . . . . . . . . . . . . . . . 22 𝑤 ∈ V
38 fveq2 6330 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑣 → (𝐹𝑥) = (𝐹𝑣))
3938breq1d 4796 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑣 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝑣)𝑆(𝐹𝑦)))
40 fveq2 6330 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
4140breq2d 4798 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑤 → ((𝐹𝑣)𝑆(𝐹𝑦) ↔ (𝐹𝑣)𝑆(𝐹𝑤)))
42 f1oweALT.1 . . . . . . . . . . . . . . . . . . . . . 22 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹𝑥)𝑆(𝐹𝑦)}
4336, 37, 39, 41, 42brab 5131 . . . . . . . . . . . . . . . . . . . . 21 (𝑣𝑅𝑤 ↔ (𝐹𝑣)𝑆(𝐹𝑤))
4435, 43syl6rbbr 279 . . . . . . . . . . . . . . . . . . . 20 ((𝑤𝑧𝑣𝑧) → (𝑣𝑅𝑤 ↔ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤)))
4544notbid 307 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝑧𝑣𝑧) → (¬ 𝑣𝑅𝑤 ↔ ¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤)))
4645ralbidva 3134 . . . . . . . . . . . . . . . . . 18 (𝑤𝑧 → (∀𝑣𝑧 ¬ 𝑣𝑅𝑤 ↔ ∀𝑣𝑧 ¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤)))
4746rexbiia 3188 . . . . . . . . . . . . . . . . 17 (∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤 ↔ ∃𝑤𝑧𝑣𝑧 ¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤))
48 breq1 4789 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹𝑧)‘𝑣) = 𝑓 → (((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤) ↔ 𝑓𝑆((𝐹𝑧)‘𝑤)))
4948notbid 307 . . . . . . . . . . . . . . . . . . . 20 (((𝐹𝑧)‘𝑣) = 𝑓 → (¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤) ↔ ¬ 𝑓𝑆((𝐹𝑧)‘𝑤)))
5049cbvfo 6685 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑧):𝑧onto→(𝐹𝑧) → (∀𝑣𝑧 ¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤) ↔ ∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆((𝐹𝑧)‘𝑤)))
5150rexbidv 3200 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑧):𝑧onto→(𝐹𝑧) → (∃𝑤𝑧𝑣𝑧 ¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤) ↔ ∃𝑤𝑧𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆((𝐹𝑧)‘𝑤)))
52 breq2 4790 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹𝑧)‘𝑤) = 𝑢 → (𝑓𝑆((𝐹𝑧)‘𝑤) ↔ 𝑓𝑆𝑢))
5352notbid 307 . . . . . . . . . . . . . . . . . . . 20 (((𝐹𝑧)‘𝑤) = 𝑢 → (¬ 𝑓𝑆((𝐹𝑧)‘𝑤) ↔ ¬ 𝑓𝑆𝑢))
5453ralbidv 3135 . . . . . . . . . . . . . . . . . . 19 (((𝐹𝑧)‘𝑤) = 𝑢 → (∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆((𝐹𝑧)‘𝑤) ↔ ∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
5554cbvexfo 6686 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑧):𝑧onto→(𝐹𝑧) → (∃𝑤𝑧𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆((𝐹𝑧)‘𝑤) ↔ ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
5651, 55bitrd 268 . . . . . . . . . . . . . . . . 17 ((𝐹𝑧):𝑧onto→(𝐹𝑧) → (∃𝑤𝑧𝑣𝑧 ¬ ((𝐹𝑧)‘𝑣)𝑆((𝐹𝑧)‘𝑤) ↔ ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
5747, 56syl5bb 272 . . . . . . . . . . . . . . . 16 ((𝐹𝑧):𝑧onto→(𝐹𝑧) → (∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤 ↔ ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
5832, 57syl 17 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → (∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤 ↔ ∃𝑢 ∈ (𝐹𝑧)∀𝑓 ∈ (𝐹𝑧) ¬ 𝑓𝑆𝑢))
5931, 58sylibrd 249 . . . . . . . . . . . . . 14 ((Fun 𝐹𝑧 ⊆ dom 𝐹) → ((𝑧 ≠ ∅ ∧ 𝑆 Fr ran 𝐹) → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤))
6059exp4b 417 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝑧 ⊆ dom 𝐹 → (𝑧 ≠ ∅ → (𝑆 Fr ran 𝐹 → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤))))
6160com34 91 . . . . . . . . . . . 12 (Fun 𝐹 → (𝑧 ⊆ dom 𝐹 → (𝑆 Fr ran 𝐹 → (𝑧 ≠ ∅ → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤))))
6261com23 86 . . . . . . . . . . 11 (Fun 𝐹 → (𝑆 Fr ran 𝐹 → (𝑧 ⊆ dom 𝐹 → (𝑧 ≠ ∅ → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤))))
6362imp4a 409 . . . . . . . . . 10 (Fun 𝐹 → (𝑆 Fr ran 𝐹 → ((𝑧 ⊆ dom 𝐹𝑧 ≠ ∅) → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤)))
6463alrimdv 2009 . . . . . . . . 9 (Fun 𝐹 → (𝑆 Fr ran 𝐹 → ∀𝑧((𝑧 ⊆ dom 𝐹𝑧 ≠ ∅) → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤)))
65 df-fr 5208 . . . . . . . . 9 (𝑅 Fr dom 𝐹 ↔ ∀𝑧((𝑧 ⊆ dom 𝐹𝑧 ≠ ∅) → ∃𝑤𝑧𝑣𝑧 ¬ 𝑣𝑅𝑤))
6664, 65syl6ibr 242 . . . . . . . 8 (Fun 𝐹 → (𝑆 Fr ran 𝐹𝑅 Fr dom 𝐹))
67 freq2 5220 . . . . . . . . 9 (dom 𝐹 = 𝐴 → (𝑅 Fr dom 𝐹𝑅 Fr 𝐴))
6867biimpd 219 . . . . . . . 8 (dom 𝐹 = 𝐴 → (𝑅 Fr dom 𝐹𝑅 Fr 𝐴))
6966, 68sylan9 497 . . . . . . 7 ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝑆 Fr ran 𝐹𝑅 Fr 𝐴))
705, 69sylbi 207 . . . . . 6 (𝐹 Fn 𝐴 → (𝑆 Fr ran 𝐹𝑅 Fr 𝐴))
714, 70sylan9r 498 . . . . 5 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
722, 71sylbi 207 . . . 4 (𝐹:𝐴onto𝐵 → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
731, 72syl 17 . . 3 (𝐹:𝐴1-1-onto𝐵 → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
74 df-f1o 6036 . . . . 5 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
75 fveq2 6330 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
7675breq1d 4796 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝑤)𝑆(𝐹𝑦)))
77 fveq2 6330 . . . . . . . . . . 11 (𝑦 = 𝑣 → (𝐹𝑦) = (𝐹𝑣))
7877breq2d 4798 . . . . . . . . . 10 (𝑦 = 𝑣 → ((𝐹𝑤)𝑆(𝐹𝑦) ↔ (𝐹𝑤)𝑆(𝐹𝑣)))
7937, 36, 76, 78, 42brab 5131 . . . . . . . . 9 (𝑤𝑅𝑣 ↔ (𝐹𝑤)𝑆(𝐹𝑣))
8079a1i 11 . . . . . . . 8 ((𝐹:𝐴1-1𝐵 ∧ (𝑤𝐴𝑣𝐴)) → (𝑤𝑅𝑣 ↔ (𝐹𝑤)𝑆(𝐹𝑣)))
81 f1fveq 6660 . . . . . . . . 9 ((𝐹:𝐴1-1𝐵 ∧ (𝑤𝐴𝑣𝐴)) → ((𝐹𝑤) = (𝐹𝑣) ↔ 𝑤 = 𝑣))
8281bicomd 213 . . . . . . . 8 ((𝐹:𝐴1-1𝐵 ∧ (𝑤𝐴𝑣𝐴)) → (𝑤 = 𝑣 ↔ (𝐹𝑤) = (𝐹𝑣)))
8343a1i 11 . . . . . . . 8 ((𝐹:𝐴1-1𝐵 ∧ (𝑤𝐴𝑣𝐴)) → (𝑣𝑅𝑤 ↔ (𝐹𝑣)𝑆(𝐹𝑤)))
8480, 82, 833orbi123d 1546 . . . . . . 7 ((𝐹:𝐴1-1𝐵 ∧ (𝑤𝐴𝑣𝐴)) → ((𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤) ↔ ((𝐹𝑤)𝑆(𝐹𝑣) ∨ (𝐹𝑤) = (𝐹𝑣) ∨ (𝐹𝑣)𝑆(𝐹𝑤))))
85842ralbidva 3137 . . . . . 6 (𝐹:𝐴1-1𝐵 → (∀𝑤𝐴𝑣𝐴 (𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤) ↔ ∀𝑤𝐴𝑣𝐴 ((𝐹𝑤)𝑆(𝐹𝑣) ∨ (𝐹𝑤) = (𝐹𝑣) ∨ (𝐹𝑣)𝑆(𝐹𝑤))))
86 breq1 4789 . . . . . . . . . 10 ((𝐹𝑤) = 𝑢 → ((𝐹𝑤)𝑆(𝐹𝑣) ↔ 𝑢𝑆(𝐹𝑣)))
87 eqeq1 2775 . . . . . . . . . 10 ((𝐹𝑤) = 𝑢 → ((𝐹𝑤) = (𝐹𝑣) ↔ 𝑢 = (𝐹𝑣)))
88 breq2 4790 . . . . . . . . . 10 ((𝐹𝑤) = 𝑢 → ((𝐹𝑣)𝑆(𝐹𝑤) ↔ (𝐹𝑣)𝑆𝑢))
8986, 87, 883orbi123d 1546 . . . . . . . . 9 ((𝐹𝑤) = 𝑢 → (((𝐹𝑤)𝑆(𝐹𝑣) ∨ (𝐹𝑤) = (𝐹𝑣) ∨ (𝐹𝑣)𝑆(𝐹𝑤)) ↔ (𝑢𝑆(𝐹𝑣) ∨ 𝑢 = (𝐹𝑣) ∨ (𝐹𝑣)𝑆𝑢)))
9089ralbidv 3135 . . . . . . . 8 ((𝐹𝑤) = 𝑢 → (∀𝑣𝐴 ((𝐹𝑤)𝑆(𝐹𝑣) ∨ (𝐹𝑤) = (𝐹𝑣) ∨ (𝐹𝑣)𝑆(𝐹𝑤)) ↔ ∀𝑣𝐴 (𝑢𝑆(𝐹𝑣) ∨ 𝑢 = (𝐹𝑣) ∨ (𝐹𝑣)𝑆𝑢)))
9190cbvfo 6685 . . . . . . 7 (𝐹:𝐴onto𝐵 → (∀𝑤𝐴𝑣𝐴 ((𝐹𝑤)𝑆(𝐹𝑣) ∨ (𝐹𝑤) = (𝐹𝑣) ∨ (𝐹𝑣)𝑆(𝐹𝑤)) ↔ ∀𝑢𝐵𝑣𝐴 (𝑢𝑆(𝐹𝑣) ∨ 𝑢 = (𝐹𝑣) ∨ (𝐹𝑣)𝑆𝑢)))
92 breq2 4790 . . . . . . . . . 10 ((𝐹𝑣) = 𝑓 → (𝑢𝑆(𝐹𝑣) ↔ 𝑢𝑆𝑓))
93 eqeq2 2782 . . . . . . . . . 10 ((𝐹𝑣) = 𝑓 → (𝑢 = (𝐹𝑣) ↔ 𝑢 = 𝑓))
94 breq1 4789 . . . . . . . . . 10 ((𝐹𝑣) = 𝑓 → ((𝐹𝑣)𝑆𝑢𝑓𝑆𝑢))
9592, 93, 943orbi123d 1546 . . . . . . . . 9 ((𝐹𝑣) = 𝑓 → ((𝑢𝑆(𝐹𝑣) ∨ 𝑢 = (𝐹𝑣) ∨ (𝐹𝑣)𝑆𝑢) ↔ (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
9695cbvfo 6685 . . . . . . . 8 (𝐹:𝐴onto𝐵 → (∀𝑣𝐴 (𝑢𝑆(𝐹𝑣) ∨ 𝑢 = (𝐹𝑣) ∨ (𝐹𝑣)𝑆𝑢) ↔ ∀𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
9796ralbidv 3135 . . . . . . 7 (𝐹:𝐴onto𝐵 → (∀𝑢𝐵𝑣𝐴 (𝑢𝑆(𝐹𝑣) ∨ 𝑢 = (𝐹𝑣) ∨ (𝐹𝑣)𝑆𝑢) ↔ ∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
9891, 97bitrd 268 . . . . . 6 (𝐹:𝐴onto𝐵 → (∀𝑤𝐴𝑣𝐴 ((𝐹𝑤)𝑆(𝐹𝑣) ∨ (𝐹𝑤) = (𝐹𝑣) ∨ (𝐹𝑣)𝑆(𝐹𝑤)) ↔ ∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
9985, 98sylan9bb 499 . . . . 5 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) → (∀𝑤𝐴𝑣𝐴 (𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤) ↔ ∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
10074, 99sylbi 207 . . . 4 (𝐹:𝐴1-1-onto𝐵 → (∀𝑤𝐴𝑣𝐴 (𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤) ↔ ∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
101100biimprd 238 . . 3 (𝐹:𝐴1-1-onto𝐵 → (∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢) → ∀𝑤𝐴𝑣𝐴 (𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤)))
10273, 101anim12d 596 . 2 (𝐹:𝐴1-1-onto𝐵 → ((𝑆 Fr 𝐵 ∧ ∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)) → (𝑅 Fr 𝐴 ∧ ∀𝑤𝐴𝑣𝐴 (𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤))))
103 dfwe2 7126 . 2 (𝑆 We 𝐵 ↔ (𝑆 Fr 𝐵 ∧ ∀𝑢𝐵𝑓𝐵 (𝑢𝑆𝑓𝑢 = 𝑓𝑓𝑆𝑢)))
104 dfwe2 7126 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑤𝐴𝑣𝐴 (𝑤𝑅𝑣𝑤 = 𝑣𝑣𝑅𝑤)))
105102, 103, 1043imtr4g 285 1 (𝐹:𝐴1-1-onto𝐵 → (𝑆 We 𝐵𝑅 We 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3o 1070  wal 1629   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  wrex 3062  Vcvv 3351  wss 3723  c0 4063   class class class wbr 4786  {copab 4846   Fr wfr 5205   We wwe 5207  dom cdm 5249  ran crn 5250  cres 5251  cima 5252  Fun wfun 6023   Fn wfn 6024  1-1wf1 6026  ontowfo 6027  1-1-ontowf1o 6028  cfv 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037
This theorem is referenced by: (None)
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