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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 44226 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 6650 | . . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝐵 → 𝐹:𝐶⟶𝐵) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) |
4 | 3 | ffvelrnda 6893 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝐹‘𝑦) ∈ 𝐵) |
5 | f1ofveu 7197 | . . . . 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 3296 | . . . . 5 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦) ↔ ∃!𝑦 ∈ 𝐶 (𝐹‘𝑦) = 𝑥) |
8 | 5, 7 | sylibr 237 | . . . 4 ⊢ ((𝐹:𝐶–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦)) |
9 | 1, 8 | sylan 583 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦)) |
10 | sbceq1a 3698 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜓 ↔ [(𝐹‘𝑦) / 𝑥]𝜓)) | |
11 | 10 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ [(𝐹‘𝑦) / 𝑥]𝜓)) |
12 | reuf1odnf.z | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜃)) | |
13 | 12 | cbvsbcvw 3722 | . . . 4 ⊢ ([(𝐹‘𝑦) / 𝑥]𝜓 ↔ [(𝐹‘𝑦) / 𝑧]𝜃) |
14 | 11, 13 | bitrdi 290 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ [(𝐹‘𝑦) / 𝑧]𝜃)) |
15 | 4, 9, 14 | reuxfr1d 3656 | . 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 226 | . . 3 ⊢ (𝜑 → ([(𝐹‘𝑦) / 𝑧]𝜃 ↔ [(𝐹‘𝑦) / 𝑥]𝜓)) |
18 | 17 | reubidv 3294 | . 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑧]𝜃 ↔ ∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑥]𝜓)) |
19 | fvexd 6721 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑦) ∈ V) | |
20 | reuf1odnf.x | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ 𝜒)) | |
21 | nfv 1922 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
22 | reuf1odnf.n | . . . . 5 ⊢ Ⅎ𝑥𝜒 | |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒) |
24 | 19, 20, 21, 23 | sbciedf 3731 | . . 3 ⊢ (𝜑 → ([(𝐹‘𝑦) / 𝑥]𝜓 ↔ 𝜒)) |
25 | 24 | reubidv 3294 | . 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐶 [(𝐹‘𝑦) / 𝑥]𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) |
26 | 15, 18, 25 | 3bitrd 308 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 ∃!wreu 3056 Vcvv 3401 [wsbc 3687 ⟶wf 6365 –1-1-onto→wf1o 6368 ‘cfv 6369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 |
This theorem is referenced by: prproropreud 44588 |
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