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Theorem reuf1odnf 47119
Description: There is exactly one element in each of two isomorphic sets. Variant of reuf1od 47120 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 6848 . . . . 5 (𝐹:𝐶1-1-onto𝐵𝐹:𝐶𝐵)
31, 2syl 17 . . . 4 (𝜑𝐹:𝐶𝐵)
43ffvelcdmda 7104 . . 3 ((𝜑𝑦𝐶) → (𝐹𝑦) ∈ 𝐵)
5 f1ofveu 7425 . . . . 5 ((𝐹:𝐶1-1-onto𝐵𝑥𝐵) → ∃!𝑦𝐶 (𝐹𝑦) = 𝑥)
6 eqcom 2744 . . . . . 6 (𝑥 = (𝐹𝑦) ↔ (𝐹𝑦) = 𝑥)
76reubii 3389 . . . . 5 (∃!𝑦𝐶 𝑥 = (𝐹𝑦) ↔ ∃!𝑦𝐶 (𝐹𝑦) = 𝑥)
85, 7sylibr 234 . . . 4 ((𝐹:𝐶1-1-onto𝐵𝑥𝐵) → ∃!𝑦𝐶 𝑥 = (𝐹𝑦))
91, 8sylan 580 . . 3 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = (𝐹𝑦))
10 sbceq1a 3799 . . . . 5 (𝑥 = (𝐹𝑦) → (𝜓[(𝐹𝑦) / 𝑥]𝜓))
1110adantl 481 . . . 4 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓[(𝐹𝑦) / 𝑥]𝜓))
12 reuf1odnf.z . . . . 5 (𝑥 = 𝑧 → (𝜓𝜃))
1312cbvsbcvw 3822 . . . 4 ([(𝐹𝑦) / 𝑥]𝜓[(𝐹𝑦) / 𝑧]𝜃)
1411, 13bitrdi 287 . . 3 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓[(𝐹𝑦) / 𝑧]𝜃))
154, 9, 14reuxfr1d 3756 . 2 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 [(𝐹𝑦) / 𝑧]𝜃))
1613a1i 11 . . . 4 (𝜑 → ([(𝐹𝑦) / 𝑥]𝜓[(𝐹𝑦) / 𝑧]𝜃))
1716bicomd 223 . . 3 (𝜑 → ([(𝐹𝑦) / 𝑧]𝜃[(𝐹𝑦) / 𝑥]𝜓))
1817reubidv 3398 . 2 (𝜑 → (∃!𝑦𝐶 [(𝐹𝑦) / 𝑧]𝜃 ↔ ∃!𝑦𝐶 [(𝐹𝑦) / 𝑥]𝜓))
19 fvexd 6921 . . . 4 (𝜑 → (𝐹𝑦) ∈ V)
20 reuf1odnf.x . . . 4 ((𝜑𝑥 = (𝐹𝑦)) → (𝜓𝜒))
21 nfv 1914 . . . 4 𝑥𝜑
22 reuf1odnf.n . . . . 5 𝑥𝜒
2322a1i 11 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
2419, 20, 21, 23sbciedf 3831 . . 3 (𝜑 → ([(𝐹𝑦) / 𝑥]𝜓𝜒))
2524reubidv 3398 . 2 (𝜑 → (∃!𝑦𝐶 [(𝐹𝑦) / 𝑥]𝜓 ↔ ∃!𝑦𝐶 𝜒))
2615, 18, 253bitrd 305 1 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wnf 1783  wcel 2108  ∃!wreu 3378  Vcvv 3480  [wsbc 3788  wf 6557  1-1-ontowf1o 6560  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  prproropreud  47496
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