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Theorem reuf1od 46517
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 6844 . . . 4 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
31, 2syl 17 . . 3 (𝜑𝐹:𝐶𝐵)
43ffvelcdmda 7099 . 2 ((𝜑𝑦𝐶) → (𝐹𝑦) ∈ 𝐵)
5 f1ofveu 7420 . . . 4 ((𝐹:𝐶1-1-onto𝐵𝑥𝐵) → ∃!𝑦𝐶 (𝐹𝑦) = 𝑥)
6 eqcom 2735 . . . . 5 (𝑥 = (𝐹𝑦) ↔ (𝐹𝑦) = 𝑥)
76reubii 3383 . . . 4 (∃!𝑦𝐶 𝑥 = (𝐹𝑦) ↔ ∃!𝑦𝐶 (𝐹𝑦) = 𝑥)
85, 7sylibr 233 . . 3 ((𝐹:𝐶1-1-onto𝐵𝑥𝐵) → ∃!𝑦𝐶 𝑥 = (𝐹𝑦))
91, 8sylan 578 . 2 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = (𝐹𝑦))
10 reuf1od.x . 2 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓𝜒))
114, 9, 10reuxfr1d 3747 1 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  ∃!wreu 3372  wf 6549  1-1-ontowf1o 6552  cfv 6553
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561
This theorem is referenced by: (None)
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