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Theorem reuf1odnf 46720
Description: There is exactly one element in each of two isomorphic sets. Variant of reuf1od 46721 with no distinct variable condition for 𝜒. (Contributed by AV, 19-Mar-2023.)
Hypotheses
Ref Expression
reuf1odnf.f (𝜑𝐹:𝐶1-1-onto𝐵)
reuf1odnf.x ((𝜑𝑥 = (𝐹𝑦)) → (𝜓𝜒))
reuf1odnf.z (𝑥 = 𝑧 → (𝜓𝜃))
reuf1odnf.n 𝑥𝜒
Assertion
Ref Expression
reuf1odnf (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦   𝜓,𝑦   𝜓,𝑧   𝜃,𝑥   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑧)   𝜓(𝑥)   𝜒(𝑥,𝑦,𝑧)   𝜃(𝑦,𝑧)   𝐵(𝑧)   𝐶(𝑧)   𝐹(𝑧)

Proof of Theorem reuf1odnf
StepHypRef Expression
1 reuf1odnf.f . . . . 5 (𝜑𝐹:𝐶1-1-onto𝐵)
2 f1of 6843 . . . . 5 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
31, 2syl 17 . . . 4 (𝜑𝐹:𝐶𝐵)
43ffvelcdmda 7098 . . 3 ((𝜑𝑦𝐶) → (𝐹𝑦) ∈ 𝐵)
5 f1ofveu 7418 . . . . 5 ((𝐹:𝐶1-1-onto𝐵𝑥𝐵) → ∃!𝑦𝐶 (𝐹𝑦) = 𝑥)
6 eqcom 2733 . . . . . 6 (𝑥 = (𝐹𝑦) ↔ (𝐹𝑦) = 𝑥)
76reubii 3373 . . . . 5 (∃!𝑦𝐶 𝑥 = (𝐹𝑦) ↔ ∃!𝑦𝐶 (𝐹𝑦) = 𝑥)
85, 7sylibr 233 . . . 4 ((𝐹:𝐶1-1-onto𝐵𝑥𝐵) → ∃!𝑦𝐶 𝑥 = (𝐹𝑦))
91, 8sylan 578 . . 3 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = (𝐹𝑦))
10 sbceq1a 3787 . . . . 5 (𝑥 = (𝐹𝑦) → (𝜓[(𝐹𝑦) / 𝑥]𝜓))
1110adantl 480 . . . 4 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓[(𝐹𝑦) / 𝑥]𝜓))
12 reuf1odnf.z . . . . 5 (𝑥 = 𝑧 → (𝜓𝜃))
1312cbvsbcvw 3811 . . . 4 ([(𝐹𝑦) / 𝑥]𝜓[(𝐹𝑦) / 𝑧]𝜃)
1411, 13bitrdi 286 . . 3 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓[(𝐹𝑦) / 𝑧]𝜃))
154, 9, 14reuxfr1d 3744 . 2 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 [(𝐹𝑦) / 𝑧]𝜃))
1613a1i 11 . . . 4 (𝜑 → ([(𝐹𝑦) / 𝑥]𝜓[(𝐹𝑦) / 𝑧]𝜃))
1716bicomd 222 . . 3 (𝜑 → ([(𝐹𝑦) / 𝑧]𝜃[(𝐹𝑦) / 𝑥]𝜓))
1817reubidv 3382 . 2 (𝜑 → (∃!𝑦𝐶 [(𝐹𝑦) / 𝑧]𝜃 ↔ ∃!𝑦𝐶 [(𝐹𝑦) / 𝑥]𝜓))
19 fvexd 6916 . . . 4 (𝜑 → (𝐹𝑦) ∈ V)
20 reuf1odnf.x . . . 4 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓𝜒))
21 nfv 1910 . . . 4 𝑥𝜑
22 reuf1odnf.n . . . . 5 𝑥𝜒
2322a1i 11 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
2419, 20, 21, 23sbciedf 3821 . . 3 (𝜑 → ([(𝐹𝑦) / 𝑥]𝜓𝜒))
2524reubidv 3382 . 2 (𝜑 → (∃!𝑦𝐶 [(𝐹𝑦) / 𝑥]𝜓 ↔ ∃!𝑦𝐶 𝜒))
2615, 18, 253bitrd 304 1 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wnf 1778  wcel 2099  ∃!wreu 3362  Vcvv 3462  [wsbc 3776  wf 6550  1-1-ontowf1o 6553  cfv 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  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 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562
This theorem is referenced by:  prproropreud  47081
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