Theorem reuf1odnf 43326

Description: There is exactly one element in each of two isomorphic sets. Variant of reuf1od 43327 with no distinct variable condition for 𝜒. (Contributed by AV, 19-Mar-2023.)

Hypotheses

Ref	Expression
reuf1odnf.f	⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵)
reuf1odnf.x	⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ 𝜒))
reuf1odnf.z	⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜃))
reuf1odnf.n	⊢ Ⅎ𝑥𝜒

Assertion

Ref	Expression
reuf1odnf	⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 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

Step	Hyp	Ref	Expression
1		reuf1odnf.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵)
2		f1of 6615	. . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝐵 → 𝐹:𝐶⟶𝐵)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵)
4	3	ffvelrnda 6851	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝐹‘𝑦) ∈ 𝐵)
5		f1ofveu 7151	. . . . 5 ⊢ ((𝐹:𝐶–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 (𝐹‘𝑦) = 𝑥)
6		eqcom 2828	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = 𝑥)
7	6	reubii 3391	. . . . 5 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦) ↔ ∃!𝑦 ∈ 𝐶 (𝐹‘𝑦) = 𝑥)
8	5, 7	sylibr 236	. . . 4 ⊢ ((𝐹:𝐶–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦))
9	1, 8	sylan 582	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦))
10		sbceq1a 3783	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜓 ↔ [(𝐹‘𝑦) / 𝑥]𝜓))
11	10	adantl 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ [(𝐹‘𝑦) / 𝑥]𝜓))
12		reuf1odnf.z	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜃))
13	12	cbvsbcvw 3805	. . . 4 ⊢ ([(𝐹‘𝑦) / 𝑥]𝜓 ↔ [(𝐹‘𝑦) / 𝑧]𝜃)
14	11, 13	syl6bb 289	. . 3 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ [(𝐹‘𝑦) / 𝑧]𝜃))
15	4, 9, 14	reuxfr1d 3741	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑧]𝜃))
16	13	a1i 11	. . . 4 ⊢ (𝜑 → ([(𝐹‘𝑦) / 𝑥]𝜓 ↔ [(𝐹‘𝑦) / 𝑧]𝜃))
17	16	bicomd 225	. . 3 ⊢ (𝜑 → ([(𝐹‘𝑦) / 𝑧]𝜃 ↔ [(𝐹‘𝑦) / 𝑥]𝜓))
18	17	reubidv 3389	. 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑧]𝜃 ↔ ∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑥]𝜓))
19		fvexd 6685	. . . 4 ⊢ (𝜑 → (𝐹‘𝑦) ∈ V)
20		reuf1odnf.x	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ 𝜒))
21		nfv 1915	. . . 4 ⊢ Ⅎ𝑥𝜑
22		reuf1odnf.n	. . . . 5 ⊢ Ⅎ𝑥𝜒
23	22	a1i 11	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
24	19, 20, 21, 23	sbciedf 3813	. . 3 ⊢ (𝜑 → ([(𝐹‘𝑦) / 𝑥]𝜓 ↔ 𝜒))
25	24	reubidv 3389	. 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑥]𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))
26	15, 18, 25	3bitrd 307	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  ∃!wreu 3140  Vcvv 3494  [wsbc 3772  ⟶wf 6351  –1-1-onto→wf1o 6354  ‘cfv 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363

This theorem is referenced by:  prproropreud  43691