Theorem reuf1od 43664

Description: There is exactly one element in each of two isomorphic sets. (Contributed by AV, 19-Mar-2023.)

Hypotheses

Ref	Expression
reuf1od.f	⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵)
reuf1od.x	⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
reuf1od	⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem reuf1od

Step	Hyp	Ref	Expression
1		reuf1od.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵)
2		f1of 6590	. . . 4 ⊢ (𝐹:𝐶–1-1-onto→𝐵 → 𝐹:𝐶⟶𝐵)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵)
4	3	ffvelrnda 6828	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝐹‘𝑦) ∈ 𝐵)
5		f1ofveu 7130	. . . 4 ⊢ ((𝐹:𝐶–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 (𝐹‘𝑦) = 𝑥)
6		eqcom 2805	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = 𝑥)
7	6	reubii 3344	. . . 4 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦) ↔ ∃!𝑦 ∈ 𝐶 (𝐹‘𝑦) = 𝑥)
8	5, 7	sylibr 237	. . 3 ⊢ ((𝐹:𝐶–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦))
9	1, 8	sylan 583	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦))
10		reuf1od.x	. 2 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ 𝜒))
11	4, 9, 10	reuxfr1d 3689	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃!wreu 3108  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332

This theorem is referenced by: (None)