![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
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 47057 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 6848 | . . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝐵 → 𝐹:𝐶⟶𝐵) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) |
4 | 3 | ffvelcdmda 7103 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝐹‘𝑦) ∈ 𝐵) |
5 | f1ofveu 7424 | . . . . 5 ⊢ ((𝐹:𝐶–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 (𝐹‘𝑦) = 𝑥) | |
6 | eqcom 2741 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = 𝑥) | |
7 | 6 | reubii 3386 | . . . . 5 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦) ↔ ∃!𝑦 ∈ 𝐶 (𝐹‘𝑦) = 𝑥) |
8 | 5, 7 | sylibr 234 | . . . 4 ⊢ ((𝐹:𝐶–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦)) |
9 | 1, 8 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦)) |
10 | sbceq1a 3801 | . . . . 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 3825 | . . . 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 3758 | . 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 223 | . . 3 ⊢ (𝜑 → ([(𝐹‘𝑦) / 𝑧]𝜃 ↔ [(𝐹‘𝑦) / 𝑥]𝜓)) |
18 | 17 | reubidv 3395 | . 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑧]𝜃 ↔ ∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑥]𝜓)) |
19 | fvexd 6921 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑦) ∈ V) | |
20 | reuf1odnf.x | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ 𝜒)) | |
21 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
22 | reuf1odnf.n | . . . . 5 ⊢ Ⅎ𝑥𝜒 | |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒) |
24 | 19, 20, 21, 23 | sbciedf 3835 | . . 3 ⊢ (𝜑 → ([(𝐹‘𝑦) / 𝑥]𝜓 ↔ 𝜒)) |
25 | 24 | reubidv 3395 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 Ⅎwnf 1779 ∈ wcel 2105 ∃!wreu 3375 Vcvv 3477 [wsbc 3790 ⟶wf 6558 –1-1-onto→wf1o 6561 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 |
This theorem is referenced by: prproropreud 47433 |
Copyright terms: Public domain | W3C validator |