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Mirrors > Home > MPE Home > Th. List > Mathboxes > reuf1odnf | Structured version Visualization version GIF version |
Description: There is exactly one element in each of two isomorphic sets. Variant of reuf1od 46721 with no distinct variable condition for 𝜒. (Contributed by AV, 19-Mar-2023.) |
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 |
Ref | Expression |
---|---|
reuf1odnf | ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuf1odnf.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵) | |
2 | f1of 6843 | . . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝐵 → 𝐹:𝐶⟶𝐵) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) |
4 | 3 | ffvelcdmda 7098 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝐹‘𝑦) ∈ 𝐵) |
5 | f1ofveu 7418 | . . . . 5 ⊢ ((𝐹:𝐶–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 (𝐹‘𝑦) = 𝑥) | |
6 | eqcom 2733 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = 𝑥) | |
7 | 6 | reubii 3373 | . . . . 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 578 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦)) |
10 | sbceq1a 3787 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜓 ↔ [(𝐹‘𝑦) / 𝑥]𝜓)) | |
11 | 10 | adantl 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ [(𝐹‘𝑦) / 𝑥]𝜓)) |
12 | reuf1odnf.z | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜃)) | |
13 | 12 | cbvsbcvw 3811 | . . . 4 ⊢ ([(𝐹‘𝑦) / 𝑥]𝜓 ↔ [(𝐹‘𝑦) / 𝑧]𝜃) |
14 | 11, 13 | bitrdi 286 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ [(𝐹‘𝑦) / 𝑧]𝜃)) |
15 | 4, 9, 14 | reuxfr1d 3744 | . 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 3382 | . 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑧]𝜃 ↔ ∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑥]𝜓)) |
19 | fvexd 6916 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑦) ∈ V) | |
20 | reuf1odnf.x | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ 𝜒)) | |
21 | nfv 1910 | . . . 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 3382 | . 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑥]𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) |
26 | 15, 18, 25 | 3bitrd 304 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ∃!wreu 3362 Vcvv 3462 [wsbc 3776 ⟶wf 6550 –1-1-onto→wf1o 6553 ‘cfv 6554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 |
This theorem is referenced by: prproropreud 47081 |
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