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Theorem fundmen 6374
 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 4646 . . 3 dom 𝐹 ∈ V
32a1i 9 . 2 (Fun 𝐹 → dom 𝐹 ∈ V)
41a1i 9 . 2 (Fun 𝐹𝐹 ∈ V)
5 funfvop 5331 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
65ex 113 . 2 (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹))
7 funrel 4969 . . 3 (Fun 𝐹 → Rel 𝐹)
8 elreldm 4608 . . . 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 4398 . . . . . . . . 9 (Rel 𝐹𝐹 ⊆ (V × V))
127, 11sylib 120 . . . . . . . 8 (Fun 𝐹𝐹 ⊆ (V × V))
1312sselda 3008 . . . . . . 7 ((Fun 𝐹𝑦𝐹) → 𝑦 ∈ (V × V))
14 elvv 4448 . . . . . . 7 (𝑦 ∈ (V × V) ↔ ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
1513, 14sylib 120 . . . . . 6 ((Fun 𝐹𝑦𝐹) → ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
16 inteq 3659 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧, 𝑤⟩)
1716inteqd 3661 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧, 𝑤⟩)
18 vex 2613 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
19 vex 2613 . . . . . . . . . . . . . . . . 17 𝑤 ∈ V
2018, 19op1stb 4255 . . . . . . . . . . . . . . . 16 𝑧, 𝑤⟩ = 𝑧
2117, 20syl6eq 2131 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧)
22 eqeq1 2089 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 = 𝑧 𝑦 = 𝑧))
2321, 22syl5ibr 154 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑥 = 𝑧))
24 opeq1 3590 . . . . . . . . . . . . . 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 2092 . . . . . . . . . . . . . 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 2151 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → (𝑦𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
3433adantl 271 . . . . . . . . . . . . . 14 ((Fun 𝐹 ∧ (𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
35 funopfv 5265 . . . . . . . . . . . . . . 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 3597 . . . . . . . . 9 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝑥, 𝑤⟩)
4232, 41eqtr4d 2118 . . . . . . . 8 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → 𝑦 = ⟨𝑥, (𝐹𝑥)⟩)
4342exp32 357 . . . . . . 7 ((Fun 𝐹𝑦𝐹) → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩)))
4443exlimdvv 1820 . . . . . 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 3659 . . . . . . . . 9 (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑦 = 𝑥, (𝐹𝑥)⟩)
4847inteqd 3661 . . . . . . . 8 (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑦 = 𝑥, (𝐹𝑥)⟩)
4948adantl 271 . . . . . . 7 (((Fun 𝐹𝑥 ∈ dom 𝐹) ∧ 𝑦 = ⟨𝑥, (𝐹𝑥)⟩) → 𝑦 = 𝑥, (𝐹𝑥)⟩)
50 vex 2613 . . . . . . . . 9 𝑥 ∈ V
51 funfvex 5243 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
52 op1stbg 4256 . . . . . . . . 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 2116 . . . . . 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 6337 1 (Fun 𝐹 → dom 𝐹𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1285  ∃wex 1422   ∈ wcel 1434  Vcvv 2610   ⊆ wss 2982  ⟨cop 3419  ∩ cint 3656   class class class wbr 3805   × cxp 4389  dom cdm 4391  Rel wrel 4396  Fun wfun 4946  ‘cfv 4952   ≈ cen 6306 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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-en 6309 This theorem is referenced by:  fundmeng  6375
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