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Theorem enerOLD 7955
Description: Obsolete proof of ener 7954 as of 1-May-2021. Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
enerOLD ≈ Er V

Proof of Theorem enerOLD
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relen 7912 . . . 4 Rel ≈
21a1i 11 . . 3 (⊤ → Rel ≈ )
3 bren 7916 . . . . 5 (𝑥𝑦 ↔ ∃𝑓 𝑓:𝑥1-1-onto𝑦)
4 f1ocnv 6111 . . . . . . 7 (𝑓:𝑥1-1-onto𝑦𝑓:𝑦1-1-onto𝑥)
5 vex 3192 . . . . . . . 8 𝑦 ∈ V
6 vex 3192 . . . . . . . 8 𝑥 ∈ V
7 f1oen2g 7924 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ 𝑓:𝑦1-1-onto𝑥) → 𝑦𝑥)
85, 6, 7mp3an12 1411 . . . . . . 7 (𝑓:𝑦1-1-onto𝑥𝑦𝑥)
94, 8syl 17 . . . . . 6 (𝑓:𝑥1-1-onto𝑦𝑦𝑥)
109exlimiv 1855 . . . . 5 (∃𝑓 𝑓:𝑥1-1-onto𝑦𝑦𝑥)
113, 10sylbi 207 . . . 4 (𝑥𝑦𝑦𝑥)
1211adantl 482 . . 3 ((⊤ ∧ 𝑥𝑦) → 𝑦𝑥)
13 bren 7916 . . . . 5 (𝑥𝑦 ↔ ∃𝑔 𝑔:𝑥1-1-onto𝑦)
14 bren 7916 . . . . 5 (𝑦𝑧 ↔ ∃𝑓 𝑓:𝑦1-1-onto𝑧)
15 eeanv 2181 . . . . . 6 (∃𝑔𝑓(𝑔:𝑥1-1-onto𝑦𝑓:𝑦1-1-onto𝑧) ↔ (∃𝑔 𝑔:𝑥1-1-onto𝑦 ∧ ∃𝑓 𝑓:𝑦1-1-onto𝑧))
16 f1oco 6121 . . . . . . . . 9 ((𝑓:𝑦1-1-onto𝑧𝑔:𝑥1-1-onto𝑦) → (𝑓𝑔):𝑥1-1-onto𝑧)
1716ancoms 469 . . . . . . . 8 ((𝑔:𝑥1-1-onto𝑦𝑓:𝑦1-1-onto𝑧) → (𝑓𝑔):𝑥1-1-onto𝑧)
18 vex 3192 . . . . . . . . 9 𝑧 ∈ V
19 f1oen2g 7924 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑧 ∈ V ∧ (𝑓𝑔):𝑥1-1-onto𝑧) → 𝑥𝑧)
206, 18, 19mp3an12 1411 . . . . . . . 8 ((𝑓𝑔):𝑥1-1-onto𝑧𝑥𝑧)
2117, 20syl 17 . . . . . . 7 ((𝑔:𝑥1-1-onto𝑦𝑓:𝑦1-1-onto𝑧) → 𝑥𝑧)
2221exlimivv 1857 . . . . . 6 (∃𝑔𝑓(𝑔:𝑥1-1-onto𝑦𝑓:𝑦1-1-onto𝑧) → 𝑥𝑧)
2315, 22sylbir 225 . . . . 5 ((∃𝑔 𝑔:𝑥1-1-onto𝑦 ∧ ∃𝑓 𝑓:𝑦1-1-onto𝑧) → 𝑥𝑧)
2413, 14, 23syl2anb 496 . . . 4 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
2524adantl 482 . . 3 ((⊤ ∧ (𝑥𝑦𝑦𝑧)) → 𝑥𝑧)
266enref 7940 . . . . 5 𝑥𝑥
276, 262th 254 . . . 4 (𝑥 ∈ V ↔ 𝑥𝑥)
2827a1i 11 . . 3 (⊤ → (𝑥 ∈ V ↔ 𝑥𝑥))
292, 12, 25, 28iserd 7720 . 2 (⊤ → ≈ Er V)
3029trud 1490 1 ≈ Er V
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wtru 1481  wex 1701  wcel 1987  Vcvv 3189   class class class wbr 4618  ccnv 5078  ccom 5083  Rel wrel 5084  1-1-ontowf1o 5851   Er wer 7691  cen 7904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-er 7694  df-en 7908
This theorem is referenced by: (None)
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