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Theorem reuf1odnf 45894
Description: There is exactly one element in each of two isomorphic sets. Variant of reuf1od 45895 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 6833 . . . . 5 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
31, 2syl 17 . . . 4 (𝜑𝐹:𝐶𝐵)
43ffvelcdmda 7086 . . 3 ((𝜑𝑦𝐶) → (𝐹𝑦) ∈ 𝐵)
5 f1ofveu 7405 . . . . 5 ((𝐹:𝐶1-1-onto𝐵𝑥𝐵) → ∃!𝑦𝐶 (𝐹𝑦) = 𝑥)
6 eqcom 2739 . . . . . 6 (𝑥 = (𝐹𝑦) ↔ (𝐹𝑦) = 𝑥)
76reubii 3385 . . . . 5 (∃!𝑦𝐶 𝑥 = (𝐹𝑦) ↔ ∃!𝑦𝐶 (𝐹𝑦) = 𝑥)
85, 7sylibr 233 . . . 4 ((𝐹:𝐶1-1-onto𝐵𝑥𝐵) → ∃!𝑦𝐶 𝑥 = (𝐹𝑦))
91, 8sylan 580 . . 3 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = (𝐹𝑦))
10 sbceq1a 3788 . . . . 5 (𝑥 = (𝐹𝑦) → (𝜓[(𝐹𝑦) / 𝑥]𝜓))
1110adantl 482 . . . 4 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓[(𝐹𝑦) / 𝑥]𝜓))
12 reuf1odnf.z . . . . 5 (𝑥 = 𝑧 → (𝜓𝜃))
1312cbvsbcvw 3812 . . . 4 ([(𝐹𝑦) / 𝑥]𝜓[(𝐹𝑦) / 𝑧]𝜃)
1411, 13bitrdi 286 . . 3 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓[(𝐹𝑦) / 𝑧]𝜃))
154, 9, 14reuxfr1d 3746 . 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 3394 . 2 (𝜑 → (∃!𝑦𝐶 [(𝐹𝑦) / 𝑧]𝜃 ↔ ∃!𝑦𝐶 [(𝐹𝑦) / 𝑥]𝜓))
19 fvexd 6906 . . . 4 (𝜑 → (𝐹𝑦) ∈ V)
20 reuf1odnf.x . . . 4 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓𝜒))
21 nfv 1917 . . . 4 𝑥𝜑
22 reuf1odnf.n . . . . 5 𝑥𝜒
2322a1i 11 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
2419, 20, 21, 23sbciedf 3821 . . 3 (𝜑 → ([(𝐹𝑦) / 𝑥]𝜓𝜒))
2524reubidv 3394 . 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 396   = wceq 1541  wnf 1785  wcel 2106  ∃!wreu 3374  Vcvv 3474  [wsbc 3777  wf 6539  1-1-ontowf1o 6542  cfv 6543
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551
This theorem is referenced by:  prproropreud  46256
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