Theorem reuf1odnf 46114

Description: There is exactly one element in each of two isomorphic sets. Variant of reuf1od 46115 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 6833	. . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝐵 → 𝐹:𝐶⟶𝐵)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵)
4	3	ffvelcdmda 7086	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝐹‘𝑦) ∈ 𝐵)
5		f1ofveu 7406	. . . . 5 ⊢ ((𝐹:𝐶–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 (𝐹‘𝑦) = 𝑥)
6		eqcom 2738	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = 𝑥)
7	6	reubii 3384	. . . . 5 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦) ↔ ∃!𝑦 ∈ 𝐶 (𝐹‘𝑦) = 𝑥)
8	5, 7	sylibr 233	. . . 4 ⊢ ((𝐹:𝐶–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦))
9	1, 8	sylan 579	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦))
10		sbceq1a 3788	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜓 ↔ [(𝐹‘𝑦) / 𝑥]𝜓))
11	10	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ [(𝐹‘𝑦) / 𝑥]𝜓))
12		reuf1odnf.z	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜃))
13	12	cbvsbcvw 3812	. . . 4 ⊢ ([(𝐹‘𝑦) / 𝑥]𝜓 ↔ [(𝐹‘𝑦) / 𝑧]𝜃)
14	11, 13	bitrdi 287	. . 3 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ [(𝐹‘𝑦) / 𝑧]𝜃))
15	4, 9, 14	reuxfr1d 3746	. 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 222	. . 3 ⊢ (𝜑 → ([(𝐹‘𝑦) / 𝑧]𝜃 ↔ [(𝐹‘𝑦) / 𝑥]𝜓))
18	17	reubidv 3393	. 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑧]𝜃 ↔ ∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑥]𝜓))
19		fvexd 6906	. . . 4 ⊢ (𝜑 → (𝐹‘𝑦) ∈ V)
20		reuf1odnf.x	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ 𝜒))
21		nfv 1916	. . . 4 ⊢ Ⅎ𝑥𝜑
22		reuf1odnf.n	. . . . 5 ⊢ Ⅎ𝑥𝜒
23	22	a1i 11	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
24	19, 20, 21, 23	sbciedf 3821	. . 3 ⊢ (𝜑 → ([(𝐹‘𝑦) / 𝑥]𝜓 ↔ 𝜒))
25	24	reubidv 3393	. 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑥]𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))
26	15, 18, 25	3bitrd 305	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540  Ⅎwnf 1784   ∈ wcel 2105  ∃!wreu 3373  Vcvv 3473  [wsbc 3777  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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  46476