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Theorem fundmen 6503
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 4687 . . 3 dom 𝐹 ∈ V
32a1i 9 . 2 (Fun 𝐹 → dom 𝐹 ∈ V)
41a1i 9 . 2 (Fun 𝐹𝐹 ∈ V)
5 funfvop 5395 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
65ex 113 . 2 (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹))
7 funrel 5019 . . 3 (Fun 𝐹 → Rel 𝐹)
8 elreldm 4649 . . . 4 ((Rel 𝐹𝑦𝐹) → 𝑦 ∈ dom 𝐹)
98ex 113 . . 3 (Rel 𝐹 → (𝑦𝐹 𝑦 ∈ dom 𝐹))
107, 9syl 14 . 2 (Fun 𝐹 → (𝑦𝐹 𝑦 ∈ dom 𝐹))
11 df-rel 4435 . . . . . . . . 9 (Rel 𝐹𝐹 ⊆ (V × V))
127, 11sylib 120 . . . . . . . 8 (Fun 𝐹𝐹 ⊆ (V × V))
1312sselda 3023 . . . . . . 7 ((Fun 𝐹𝑦𝐹) → 𝑦 ∈ (V × V))
14 elvv 4488 . . . . . . 7 (𝑦 ∈ (V × V) ↔ ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
1513, 14sylib 120 . . . . . 6 ((Fun 𝐹𝑦𝐹) → ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
16 inteq 3686 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧, 𝑤⟩)
1716inteqd 3688 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧, 𝑤⟩)
18 vex 2622 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
19 vex 2622 . . . . . . . . . . . . . . . . 17 𝑤 ∈ V
2018, 19op1stb 4290 . . . . . . . . . . . . . . . 16 𝑧, 𝑤⟩ = 𝑧
2117, 20syl6eq 2136 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧)
22 eqeq1 2094 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 = 𝑧 𝑦 = 𝑧))
2321, 22syl5ibr 154 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑥 = 𝑧))
24 opeq1 3617 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
2523, 24syl6 33 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩))
2625imp 122 . . . . . . . . . . . 12 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
27 eqeq2 2097 . . . . . . . . . . . . . 14 (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → (𝑦 = ⟨𝑥, 𝑤⟩ ↔ 𝑦 = ⟨𝑧, 𝑤⟩))
2827biimprcd 158 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
2928adantl 271 . . . . . . . . . . . 12 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
3026, 29mpd 13 . . . . . . . . . . 11 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → 𝑦 = ⟨𝑥, 𝑤⟩)
3130ancoms 264 . . . . . . . . . 10 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦) → 𝑦 = ⟨𝑥, 𝑤⟩)
3231adantl 271 . . . . . . . . 9 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → 𝑦 = ⟨𝑥, 𝑤⟩)
3330eleq1d 2156 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → (𝑦𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
3433adantl 271 . . . . . . . . . . . . . 14 ((Fun 𝐹 ∧ (𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
35 funopfv 5328 . . . . . . . . . . . . . . 15 (Fun 𝐹 → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹𝑥) = 𝑤))
3635adantr 270 . . . . . . . . . . . . . 14 ((Fun 𝐹 ∧ (𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩)) → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹𝑥) = 𝑤))
3734, 36sylbid 148 . . . . . . . . . . . . 13 ((Fun 𝐹 ∧ (𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦𝐹 → (𝐹𝑥) = 𝑤))
3837exp32 357 . . . . . . . . . . . 12 (Fun 𝐹 → (𝑥 = 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹 → (𝐹𝑥) = 𝑤))))
3938com24 86 . . . . . . . . . . 11 (Fun 𝐹 → (𝑦𝐹 → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = 𝑦 → (𝐹𝑥) = 𝑤))))
4039imp43 347 . . . . . . . . . 10 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → (𝐹𝑥) = 𝑤)
4140opeq2d 3624 . . . . . . . . 9 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝑥, 𝑤⟩)
4232, 41eqtr4d 2123 . . . . . . . 8 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → 𝑦 = ⟨𝑥, (𝐹𝑥)⟩)
4342exp32 357 . . . . . . 7 ((Fun 𝐹𝑦𝐹) → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩)))
4443exlimdvv 1825 . . . . . 6 ((Fun 𝐹𝑦𝐹) → (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩)))
4515, 44mpd 13 . . . . 5 ((Fun 𝐹𝑦𝐹) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩))
4645adantrl 462 . . . 4 ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹𝑦𝐹)) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩))
47 inteq 3686 . . . . . . . . 9 (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑦 = 𝑥, (𝐹𝑥)⟩)
4847inteqd 3688 . . . . . . . 8 (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑦 = 𝑥, (𝐹𝑥)⟩)
4948adantl 271 . . . . . . 7 (((Fun 𝐹𝑥 ∈ dom 𝐹) ∧ 𝑦 = ⟨𝑥, (𝐹𝑥)⟩) → 𝑦 = 𝑥, (𝐹𝑥)⟩)
50 vex 2622 . . . . . . . . 9 𝑥 ∈ V
51 funfvex 5306 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
52 op1stbg 4291 . . . . . . . . 9 ((𝑥 ∈ V ∧ (𝐹𝑥) ∈ V) → 𝑥, (𝐹𝑥)⟩ = 𝑥)
5350, 51, 52sylancr 405 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → 𝑥, (𝐹𝑥)⟩ = 𝑥)
5453adantr 270 . . . . . . 7 (((Fun 𝐹𝑥 ∈ dom 𝐹) ∧ 𝑦 = ⟨𝑥, (𝐹𝑥)⟩) → 𝑥, (𝐹𝑥)⟩ = 𝑥)
5549, 54eqtr2d 2121 . . . . . 6 (((Fun 𝐹𝑥 ∈ dom 𝐹) ∧ 𝑦 = ⟨𝑥, (𝐹𝑥)⟩) → 𝑥 = 𝑦)
5655ex 113 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑥 = 𝑦))
5756adantrr 463 . . . 4 ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹𝑦𝐹)) → (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑥 = 𝑦))
5846, 57impbid 127 . . 3 ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹𝑦𝐹)) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩))
5958ex 113 . 2 (Fun 𝐹 → ((𝑥 ∈ dom 𝐹𝑦𝐹) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩)))
603, 4, 6, 10, 59en3d 6466 1 (Fun 𝐹 → dom 𝐹𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  wss 2997  cop 3444   cint 3683   class class class wbr 3837   × cxp 4426  dom cdm 4428  Rel wrel 4433  Fun wfun 4996  cfv 5002  cen 6435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-en 6438
This theorem is referenced by:  fundmeng  6504
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