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Theorem fundmen 8372
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 7425 . . 3 dom 𝐹 ∈ V
32a1i 11 . 2 (Fun 𝐹 → dom 𝐹 ∈ V)
41a1i 11 . 2 (Fun 𝐹𝐹 ∈ V)
5 funfvop 6639 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
65ex 405 . 2 (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹))
7 funrel 6199 . . 3 (Fun 𝐹 → Rel 𝐹)
8 elreldm 5641 . . . 4 ((Rel 𝐹𝑦𝐹) → 𝑦 ∈ dom 𝐹)
98ex 405 . . 3 (Rel 𝐹 → (𝑦𝐹 𝑦 ∈ dom 𝐹))
107, 9syl 17 . 2 (Fun 𝐹 → (𝑦𝐹 𝑦 ∈ dom 𝐹))
11 df-rel 5407 . . . . . . . . 9 (Rel 𝐹𝐹 ⊆ (V × V))
127, 11sylib 210 . . . . . . . 8 (Fun 𝐹𝐹 ⊆ (V × V))
1312sselda 3854 . . . . . . 7 ((Fun 𝐹𝑦𝐹) → 𝑦 ∈ (V × V))
14 elvv 5469 . . . . . . 7 (𝑦 ∈ (V × V) ↔ ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
1513, 14sylib 210 . . . . . 6 ((Fun 𝐹𝑦𝐹) → ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
16 inteq 4746 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧, 𝑤⟩)
1716inteqd 4748 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧, 𝑤⟩)
18 vex 3412 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
19 vex 3412 . . . . . . . . . . . . . . . . 17 𝑤 ∈ V
2018, 19op1stb 5213 . . . . . . . . . . . . . . . 16 𝑧, 𝑤⟩ = 𝑧
2117, 20syl6eq 2824 . . . . . . . . . . . . . . 15 (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑦 = 𝑧)
22 eqeq1 2776 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 = 𝑧 𝑦 = 𝑧))
2321, 22syl5ibr 238 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑥 = 𝑧))
24 opeq1 4671 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
2523, 24syl6 35 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩))
2625imp 398 . . . . . . . . . . . 12 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
27 eqeq2 2783 . . . . . . . . . . . . . 14 (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → (𝑦 = ⟨𝑥, 𝑤⟩ ↔ 𝑦 = ⟨𝑧, 𝑤⟩))
2827biimprcd 242 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
2928adantl 474 . . . . . . . . . . . 12 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
3026, 29mpd 15 . . . . . . . . . . 11 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → 𝑦 = ⟨𝑥, 𝑤⟩)
3130ancoms 451 . . . . . . . . . 10 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦) → 𝑦 = ⟨𝑥, 𝑤⟩)
3231adantl 474 . . . . . . . . 9 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → 𝑦 = ⟨𝑥, 𝑤⟩)
3330eleq1d 2844 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩) → (𝑦𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
3433adantl 474 . . . . . . . . . . . . . 14 ((Fun 𝐹 ∧ (𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
35 funopfv 6541 . . . . . . . . . . . . . . 15 (Fun 𝐹 → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹𝑥) = 𝑤))
3635adantr 473 . . . . . . . . . . . . . 14 ((Fun 𝐹 ∧ (𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩)) → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹𝑥) = 𝑤))
3734, 36sylbid 232 . . . . . . . . . . . . 13 ((Fun 𝐹 ∧ (𝑥 = 𝑦𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦𝐹 → (𝐹𝑥) = 𝑤))
3837exp32 413 . . . . . . . . . . . 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 420 . . . . . . . . . 10 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → (𝐹𝑥) = 𝑤)
4140opeq2d 4678 . . . . . . . . 9 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝑥, 𝑤⟩)
4232, 41eqtr4d 2811 . . . . . . . 8 (((Fun 𝐹𝑦𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = 𝑦)) → 𝑦 = ⟨𝑥, (𝐹𝑥)⟩)
4342exp32 413 . . . . . . 7 ((Fun 𝐹𝑦𝐹) → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩)))
4443exlimdvv 1893 . . . . . 6 ((Fun 𝐹𝑦𝐹) → (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩)))
4515, 44mpd 15 . . . . 5 ((Fun 𝐹𝑦𝐹) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩))
4645adantrl 703 . . . 4 ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹𝑦𝐹)) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩))
47 inteq 4746 . . . . . 6 (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑦 = 𝑥, (𝐹𝑥)⟩)
4847inteqd 4748 . . . . 5 (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑦 = 𝑥, (𝐹𝑥)⟩)
49 vex 3412 . . . . . 6 𝑥 ∈ V
50 fvex 6506 . . . . . 6 (𝐹𝑥) ∈ V
5149, 50op1stb 5213 . . . . 5 𝑥, (𝐹𝑥)⟩ = 𝑥
5248, 51syl6req 2825 . . . 4 (𝑦 = ⟨𝑥, (𝐹𝑥)⟩ → 𝑥 = 𝑦)
5346, 52impbid1 217 . . 3 ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹𝑦𝐹)) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩))
5453ex 405 . 2 (Fun 𝐹 → ((𝑥 ∈ dom 𝐹𝑦𝐹) → (𝑥 = 𝑦𝑦 = ⟨𝑥, (𝐹𝑥)⟩)))
553, 4, 6, 10, 54en3d 8335 1 (Fun 𝐹 → dom 𝐹𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wex 1742  wcel 2048  Vcvv 3409  wss 3825  cop 4441   cint 4743   class class class wbr 4923   × cxp 5398  dom cdm 5400  Rel wrel 5405  Fun wfun 6176  cfv 6182  cen 8295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-int 4744  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-en 8299
This theorem is referenced by:  fundmeng  8373  infmap2  9430  heicant  34316
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