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Theorem fundmen 8266
Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypothesis
Ref Expression
fundmen.1 𝐹 ∈ V
Assertion
Ref Expression
fundmen (Fun 𝐹 → dom 𝐹𝐹)

Proof of Theorem fundmen
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fundmen.1 . . . 4 𝐹 ∈ V
21dmex 7329 . . 3 dom 𝐹 ∈ V
32a1i 11 . 2 (Fun 𝐹 → dom 𝐹 ∈ V)
41a1i 11 . 2 (Fun 𝐹𝐹 ∈ V)
5 funfvop 6551 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
65ex 399 . 2 (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹))
7 funrel 6118 . . 3 (Fun 𝐹 → Rel 𝐹)
8 elreldm 5551 . . . 4 ((Rel 𝐹𝑦𝐹) → 𝑦 ∈ dom 𝐹)
98ex 399 . . 3 (Rel 𝐹 → (𝑦𝐹 𝑦 ∈ dom 𝐹))
107, 9syl 17 . 2 (Fun 𝐹 → (𝑦𝐹 𝑦 ∈ dom 𝐹))
11 df-rel 5318 . . . . . . . . 9 (Rel 𝐹𝐹 ⊆ (V × V))
127, 11sylib 209 . . . . . . . 8 (Fun 𝐹𝐹 ⊆ (V × V))
1312sselda 3798 . . . . . . 7 ((Fun 𝐹𝑦𝐹) → 𝑦 ∈ (V × V))
14 elvv 5377 . . . . . . 7 (𝑦 ∈ (V × V) ↔ ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
1513, 14sylib 209 . . . . . 6 ((Fun 𝐹𝑦𝐹) → ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
16 inteq 4672 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧, 𝑤⟩)
1716inteqd 4674 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧, 𝑤⟩)
18 vex 3394 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
19 vex 3394 . . . . . . . . . . . . . . . . 17 𝑤 ∈ V
2018, 19op1stb 5129 . . . . . . . . . . . . . . . 16 𝑧, 𝑤⟩ = 𝑧
2117, 20syl6eq 2856 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧)
22 eqeq1 2810 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 = 𝑧 𝑦 = 𝑧))
2321, 22syl5ibr 237 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑥 = 𝑧))
24 opeq1 4595 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
2523, 24syl6 35 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩))
2625imp 395 . . . . . . . . . . . 12 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
27 eqeq2 2817 . . . . . . . . . . . . . 14 (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → (𝑦 = ⟨𝑥, 𝑤⟩ ↔ 𝑦 = ⟨𝑧, 𝑤⟩))
2827biimprcd 241 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
2928adantl 469 . . . . . . . . . . . 12 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
3026, 29mpd 15 . . . . . . . . . . 11 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → 𝑦 = ⟨𝑥, 𝑤⟩)
3130ancoms 448 . . . . . . . . . 10 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦) → 𝑦 = ⟨𝑥, 𝑤⟩)
3231adantl 469 . . . . . . . . 9 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → 𝑦 = ⟨𝑥, 𝑤⟩)
3330eleq1d 2870 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → (𝑦𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
3433adantl 469 . . . . . . . . . . . . . 14 ((Fun 𝐹 ∧ (𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
35 funopfv 6455 . . . . . . . . . . . . . . 15 (Fun 𝐹 → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹𝑥) = 𝑤))
3635adantr 468 . . . . . . . . . . . . . 14 ((Fun 𝐹 ∧ (𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩)) → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹𝑥) = 𝑤))
3734, 36sylbid 231 . . . . . . . . . . . . 13 ((Fun 𝐹 ∧ (𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦𝐹 → (𝐹𝑥) = 𝑤))
3837exp32 409 . . . . . . . . . . . 12 (Fun 𝐹 → (𝑥 = 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹 → (𝐹𝑥) = 𝑤))))
3938com24 95 . . . . . . . . . . 11 (Fun 𝐹 → (𝑦𝐹 → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = 𝑦 → (𝐹𝑥) = 𝑤))))
4039imp43 416 . . . . . . . . . 10 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → (𝐹𝑥) = 𝑤)
4140opeq2d 4602 . . . . . . . . 9 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝑥, 𝑤⟩)
4232, 41eqtr4d 2843 . . . . . . . 8 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → 𝑦 = ⟨𝑥, (𝐹𝑥)⟩)
4342exp32 409 . . . . . . 7 ((Fun 𝐹𝑦𝐹) → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩)))
4443exlimdvv 2025 . . . . . 6 ((Fun 𝐹𝑦𝐹) → (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩)))
4515, 44mpd 15 . . . . 5 ((Fun 𝐹𝑦𝐹) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩))
4645adantrl 698 . . . 4 ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹𝑦𝐹)) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩))
47 inteq 4672 . . . . . 6 (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑦 = 𝑥, (𝐹𝑥)⟩)
4847inteqd 4674 . . . . 5 (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑦 = 𝑥, (𝐹𝑥)⟩)
49 vex 3394 . . . . . 6 𝑥 ∈ V
50 fvex 6421 . . . . . 6 (𝐹𝑥) ∈ V
5149, 50op1stb 5129 . . . . 5 𝑥, (𝐹𝑥)⟩ = 𝑥
5248, 51syl6req 2857 . . . 4 (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑥 = 𝑦)
5346, 52impbid1 216 . . 3 ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹𝑦𝐹)) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩))
5453ex 399 . 2 (Fun 𝐹 → ((𝑥 ∈ dom 𝐹𝑦𝐹) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩)))
553, 4, 6, 10, 54en3d 8229 1 (Fun 𝐹 → dom 𝐹𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wex 1859  wcel 2156  Vcvv 3391  wss 3769  cop 4376   cint 4669   class class class wbr 4844   × cxp 5309  dom cdm 5311  Rel wrel 5316  Fun wfun 6095  cfv 6101  cen 8189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-int 4670  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-en 8193
This theorem is referenced by:  fundmeng  8267  infmap2  9325  heicant  33757
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