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Theorem reuf1odnf 43660
Description: There is exactly one element in each of two isomorphic sets. Variant of reuf1od 43661 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 6594 . . . . 5 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
31, 2syl 17 . . . 4 (𝜑𝐹:𝐶𝐵)
43ffvelrnda 6832 . . 3 ((𝜑𝑦𝐶) → (𝐹𝑦) ∈ 𝐵)
5 f1ofveu 7134 . . . . 5 ((𝐹:𝐶1-1-onto𝐵𝑥𝐵) → ∃!𝑦𝐶 (𝐹𝑦) = 𝑥)
6 eqcom 2808 . . . . . 6 (𝑥 = (𝐹𝑦) ↔ (𝐹𝑦) = 𝑥)
76reubii 3347 . . . . 5 (∃!𝑦𝐶 𝑥 = (𝐹𝑦) ↔ ∃!𝑦𝐶 (𝐹𝑦) = 𝑥)
85, 7sylibr 237 . . . 4 ((𝐹:𝐶1-1-onto𝐵𝑥𝐵) → ∃!𝑦𝐶 𝑥 = (𝐹𝑦))
91, 8sylan 583 . . 3 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = (𝐹𝑦))
10 sbceq1a 3734 . . . . 5 (𝑥 = (𝐹𝑦) → (𝜓[(𝐹𝑦) / 𝑥]𝜓))
1110adantl 485 . . . 4 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓[(𝐹𝑦) / 𝑥]𝜓))
12 reuf1odnf.z . . . . 5 (𝑥 = 𝑧 → (𝜓𝜃))
1312cbvsbcvw 3756 . . . 4 ([(𝐹𝑦) / 𝑥]𝜓[(𝐹𝑦) / 𝑧]𝜃)
1411, 13syl6bb 290 . . 3 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓[(𝐹𝑦) / 𝑧]𝜃))
154, 9, 14reuxfr1d 3692 . 2 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 [(𝐹𝑦) / 𝑧]𝜃))
1613a1i 11 . . . 4 (𝜑 → ([(𝐹𝑦) / 𝑥]𝜓[(𝐹𝑦) / 𝑧]𝜃))
1716bicomd 226 . . 3 (𝜑 → ([(𝐹𝑦) / 𝑧]𝜃[(𝐹𝑦) / 𝑥]𝜓))
1817reubidv 3345 . 2 (𝜑 → (∃!𝑦𝐶 [(𝐹𝑦) / 𝑧]𝜃 ↔ ∃!𝑦𝐶 [(𝐹𝑦) / 𝑥]𝜓))
19 fvexd 6664 . . . 4 (𝜑 → (𝐹𝑦) ∈ V)
20 reuf1odnf.x . . . 4 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓𝜒))
21 nfv 1915 . . . 4 𝑥𝜑
22 reuf1odnf.n . . . . 5 𝑥𝜒
2322a1i 11 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
2419, 20, 21, 23sbciedf 3764 . . 3 (𝜑 → ([(𝐹𝑦) / 𝑥]𝜓𝜒))
2524reubidv 3345 . 2 (𝜑 → (∃!𝑦𝐶 [(𝐹𝑦) / 𝑥]𝜓 ↔ ∃!𝑦𝐶 𝜒))
2615, 18, 253bitrd 308 1 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wnf 1785  wcel 2112  ∃!wreu 3111  Vcvv 3444  [wsbc 3723  wf 6324  1-1-ontowf1o 6327  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336
This theorem is referenced by:  prproropreud  44023
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