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Theorem enerOLD 7889
 Description: Obsolete proof of ener 7888 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 7846 . . . 4 Rel ≈
21a1i 11 . . 3 (⊤ → Rel ≈ )
3 bren 7850 . . . . 5 (𝑥𝑦 ↔ ∃𝑓 𝑓:𝑥1-1-onto𝑦)
4 f1ocnv 6062 . . . . . . 7 (𝑓:𝑥1-1-onto𝑦𝑓:𝑦1-1-onto𝑥)
5 vex 3176 . . . . . . . 8 𝑦 ∈ V
6 vex 3176 . . . . . . . 8 𝑥 ∈ V
7 f1oen2g 7858 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ 𝑓:𝑦1-1-onto𝑥) → 𝑦𝑥)
85, 6, 7mp3an12 1406 . . . . . . 7 (𝑓:𝑦1-1-onto𝑥𝑦𝑥)
94, 8syl 17 . . . . . 6 (𝑓:𝑥1-1-onto𝑦𝑦𝑥)
109exlimiv 1845 . . . . 5 (∃𝑓 𝑓:𝑥1-1-onto𝑦𝑦𝑥)
113, 10sylbi 206 . . . 4 (𝑥𝑦𝑦𝑥)
1211adantl 481 . . 3 ((⊤ ∧ 𝑥𝑦) → 𝑦𝑥)
13 bren 7850 . . . . 5 (𝑥𝑦 ↔ ∃𝑔 𝑔:𝑥1-1-onto𝑦)
14 bren 7850 . . . . 5 (𝑦𝑧 ↔ ∃𝑓 𝑓:𝑦1-1-onto𝑧)
15 eeanv 2170 . . . . . 6 (∃𝑔𝑓(𝑔:𝑥1-1-onto𝑦𝑓:𝑦1-1-onto𝑧) ↔ (∃𝑔 𝑔:𝑥1-1-onto𝑦 ∧ ∃𝑓 𝑓:𝑦1-1-onto𝑧))
16 f1oco 6072 . . . . . . . . 9 ((𝑓:𝑦1-1-onto𝑧𝑔:𝑥1-1-onto𝑦) → (𝑓𝑔):𝑥1-1-onto𝑧)
1716ancoms 468 . . . . . . . 8 ((𝑔:𝑥1-1-onto𝑦𝑓:𝑦1-1-onto𝑧) → (𝑓𝑔):𝑥1-1-onto𝑧)
18 vex 3176 . . . . . . . . 9 𝑧 ∈ V
19 f1oen2g 7858 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑧 ∈ V ∧ (𝑓𝑔):𝑥1-1-onto𝑧) → 𝑥𝑧)
206, 18, 19mp3an12 1406 . . . . . . . 8 ((𝑓𝑔):𝑥1-1-onto𝑧𝑥𝑧)
2117, 20syl 17 . . . . . . 7 ((𝑔:𝑥1-1-onto𝑦𝑓:𝑦1-1-onto𝑧) → 𝑥𝑧)
2221exlimivv 1847 . . . . . 6 (∃𝑔𝑓(𝑔:𝑥1-1-onto𝑦𝑓:𝑦1-1-onto𝑧) → 𝑥𝑧)
2315, 22sylbir 224 . . . . 5 ((∃𝑔 𝑔:𝑥1-1-onto𝑦 ∧ ∃𝑓 𝑓:𝑦1-1-onto𝑧) → 𝑥𝑧)
2413, 14, 23syl2anb 495 . . . 4 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
2524adantl 481 . . 3 ((⊤ ∧ (𝑥𝑦𝑦𝑧)) → 𝑥𝑧)
266enref 7874 . . . . 5 𝑥𝑥
276, 262th 253 . . . 4 (𝑥 ∈ V ↔ 𝑥𝑥)
2827a1i 11 . . 3 (⊤ → (𝑥 ∈ V ↔ 𝑥𝑥))
292, 12, 25, 28iserd 7655 . 2 (⊤ → ≈ Er V)
3029trud 1484 1 ≈ Er V
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ⊤wtru 1476  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  ◡ccnv 5037   ∘ ccom 5042  Rel wrel 5043  –1-1-onto→wf1o 5803   Er wer 7626   ≈ cen 7838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-er 7629  df-en 7842 This theorem is referenced by: (None)
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