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Theorem reuf1od 46370
Description: There is exactly one element in each of two isomorphic sets. (Contributed by AV, 19-Mar-2023.)
Hypotheses
Ref Expression
reuf1od.f (𝜑𝐹:𝐶1-1-onto𝐵)
reuf1od.x ((𝜑𝑥 = (𝐹𝑦)) → (𝜓𝜒))
Assertion
Ref Expression
reuf1od (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem reuf1od
StepHypRef Expression
1 reuf1od.f . . . 4 (𝜑𝐹:𝐶1-1-onto𝐵)
2 f1of 6826 . . . 4 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
31, 2syl 17 . . 3 (𝜑𝐹:𝐶𝐵)
43ffvelcdmda 7079 . 2 ((𝜑𝑦𝐶) → (𝐹𝑦) ∈ 𝐵)
5 f1ofveu 7398 . . . 4 ((𝐹:𝐶1-1-onto𝐵𝑥𝐵) → ∃!𝑦𝐶 (𝐹𝑦) = 𝑥)
6 eqcom 2733 . . . . 5 (𝑥 = (𝐹𝑦) ↔ (𝐹𝑦) = 𝑥)
76reubii 3379 . . . 4 (∃!𝑦𝐶 𝑥 = (𝐹𝑦) ↔ ∃!𝑦𝐶 (𝐹𝑦) = 𝑥)
85, 7sylibr 233 . . 3 ((𝐹:𝐶1-1-onto𝐵𝑥𝐵) → ∃!𝑦𝐶 𝑥 = (𝐹𝑦))
91, 8sylan 579 . 2 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = (𝐹𝑦))
10 reuf1od.x . 2 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓𝜒))
114, 9, 10reuxfr1d 3741 1 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  ∃!wreu 3368  wf 6532  1-1-ontowf1o 6535  cfv 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544
This theorem is referenced by: (None)
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