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Mirrors > Home > MPE Home > Th. List > Mathboxes > reuf1od | Structured version Visualization version GIF version |
Description: There is exactly one element in each of two isomorphic sets. (Contributed by AV, 19-Mar-2023.) |
Ref | Expression |
---|---|
reuf1od.f | ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵) |
reuf1od.x | ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
reuf1od | ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuf1od.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵) | |
2 | f1of 6608 | . . . 4 ⊢ (𝐹:𝐶–1-1-onto→𝐵 → 𝐹:𝐶⟶𝐵) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) |
4 | 3 | ffvelrnda 6844 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝐹‘𝑦) ∈ 𝐵) |
5 | f1ofveu 7143 | . . . 4 ⊢ ((𝐹:𝐶–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 (𝐹‘𝑦) = 𝑥) | |
6 | eqcom 2826 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = 𝑥) | |
7 | 6 | reubii 3390 | . . . 4 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦) ↔ ∃!𝑦 ∈ 𝐶 (𝐹‘𝑦) = 𝑥) |
8 | 5, 7 | sylibr 236 | . . 3 ⊢ ((𝐹:𝐶–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦)) |
9 | 1, 8 | sylan 582 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = (𝐹‘𝑦)) |
10 | reuf1od.x | . 2 ⊢ ((𝜑 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜓 ↔ 𝜒)) | |
11 | 4, 9, 10 | reuxfr1d 3739 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∃!wreu 3138 ⟶wf 6344 –1-1-onto→wf1o 6347 ‘cfv 6348 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 |
This theorem is referenced by: (None) |
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