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Theorem fundmen 8874
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 7804 . . 3 dom 𝐹 ∈ V
32a1i 11 . 2 (Fun 𝐹 → dom 𝐹 ∈ V)
41a1i 11 . 2 (Fun 𝐹𝐹 ∈ V)
5 funfvop 6966 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
65ex 413 . 2 (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹))
7 funrel 6487 . . 3 (Fun 𝐹 → Rel 𝐹)
8 elreldm 5863 . . . 4 ((Rel 𝐹𝑦𝐹) → 𝑦 ∈ dom 𝐹)
98ex 413 . . 3 (Rel 𝐹 → (𝑦𝐹 𝑦 ∈ dom 𝐹))
107, 9syl 17 . 2 (Fun 𝐹 → (𝑦𝐹 𝑦 ∈ dom 𝐹))
11 df-rel 5614 . . . . . . . . 9 (Rel 𝐹𝐹 ⊆ (V × V))
127, 11sylib 217 . . . . . . . 8 (Fun 𝐹𝐹 ⊆ (V × V))
1312sselda 3930 . . . . . . 7 ((Fun 𝐹𝑦𝐹) → 𝑦 ∈ (V × V))
14 elvv 5679 . . . . . . 7 (𝑦 ∈ (V × V) ↔ ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
1513, 14sylib 217 . . . . . 6 ((Fun 𝐹𝑦𝐹) → ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
16 inteq 4894 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧, 𝑤⟩)
1716inteqd 4896 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧, 𝑤⟩)
18 vex 3444 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
19 vex 3444 . . . . . . . . . . . . . . . . 17 𝑤 ∈ V
2018, 19op1stb 5404 . . . . . . . . . . . . . . . 16 𝑧, 𝑤⟩ = 𝑧
2117, 20eqtrdi 2792 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧)
22 eqeq1 2740 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 = 𝑧 𝑦 = 𝑧))
2321, 22syl5ibr 245 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑥 = 𝑧))
24 opeq1 4814 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
2523, 24syl6 35 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩))
2625imp 407 . . . . . . . . . . . 12 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
27 eqeq2 2748 . . . . . . . . . . . . . 14 (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → (𝑦 = ⟨𝑥, 𝑤⟩ ↔ 𝑦 = ⟨𝑧, 𝑤⟩))
2827biimprcd 249 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
2928adantl 482 . . . . . . . . . . . 12 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
3026, 29mpd 15 . . . . . . . . . . 11 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → 𝑦 = ⟨𝑥, 𝑤⟩)
3130ancoms 459 . . . . . . . . . 10 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦) → 𝑦 = ⟨𝑥, 𝑤⟩)
3231adantl 482 . . . . . . . . 9 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → 𝑦 = ⟨𝑥, 𝑤⟩)
3330eleq1d 2821 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → (𝑦𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
3433adantl 482 . . . . . . . . . . . . . 14 ((Fun 𝐹 ∧ (𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
35 funopfv 6860 . . . . . . . . . . . . . . 15 (Fun 𝐹 → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹𝑥) = 𝑤))
3635adantr 481 . . . . . . . . . . . . . 14 ((Fun 𝐹 ∧ (𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩)) → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹𝑥) = 𝑤))
3734, 36sylbid 239 . . . . . . . . . . . . 13 ((Fun 𝐹 ∧ (𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦𝐹 → (𝐹𝑥) = 𝑤))
3837exp32 421 . . . . . . . . . . . 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 428 . . . . . . . . . 10 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → (𝐹𝑥) = 𝑤)
4140opeq2d 4821 . . . . . . . . 9 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝑥, 𝑤⟩)
4232, 41eqtr4d 2779 . . . . . . . 8 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → 𝑦 = ⟨𝑥, (𝐹𝑥)⟩)
4342exp32 421 . . . . . . 7 ((Fun 𝐹𝑦𝐹) → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩)))
4443exlimdvv 1936 . . . . . 6 ((Fun 𝐹𝑦𝐹) → (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩)))
4515, 44mpd 15 . . . . 5 ((Fun 𝐹𝑦𝐹) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩))
4645adantrl 713 . . . 4 ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹𝑦𝐹)) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩))
47 inteq 4894 . . . . . 6 (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑦 = 𝑥, (𝐹𝑥)⟩)
4847inteqd 4896 . . . . 5 (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑦 = 𝑥, (𝐹𝑥)⟩)
49 vex 3444 . . . . . 6 𝑥 ∈ V
50 fvex 6824 . . . . . 6 (𝐹𝑥) ∈ V
5149, 50op1stb 5404 . . . . 5 𝑥, (𝐹𝑥)⟩ = 𝑥
5248, 51eqtr2di 2793 . . . 4 (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑥 = 𝑦)
5346, 52impbid1 224 . . 3 ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹𝑦𝐹)) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩))
5453ex 413 . 2 (Fun 𝐹 → ((𝑥 ∈ dom 𝐹𝑦𝐹) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩)))
553, 4, 6, 10, 54en3d 8828 1 (Fun 𝐹 → dom 𝐹𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wex 1780  wcel 2105  Vcvv 3440  wss 3896  cop 4576   cint 4891   class class class wbr 5086   × cxp 5605  dom cdm 5607  Rel wrel 5612  Fun wfun 6459  cfv 6465  cen 8779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3442  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-int 4892  df-br 5087  df-opab 5149  df-mpt 5170  df-id 5506  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-en 8783
This theorem is referenced by:  fundmeng  8875  infmap2  10053  heicant  35889
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