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Mirrors > Home > MPE Home > Th. List > f1ofveu | Structured version Visualization version GIF version |
Description: There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.) |
Ref | Expression |
---|---|
f1ofveu | ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnv 6861 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
2 | f1of 6849 | . . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵⟶𝐴) |
4 | feu 6785 | . . 3 ⊢ ((◡𝐹:𝐵⟶𝐴 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 〈𝐶, 𝑥〉 ∈ ◡𝐹) | |
5 | 3, 4 | sylan 580 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 〈𝐶, 𝑥〉 ∈ ◡𝐹) |
6 | f1ocnvfvb 7299 | . . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥)) | |
7 | 6 | 3com23 1125 | . . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥)) |
8 | dff1o4 6857 | . . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) | |
9 | 8 | simprbi 496 | . . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹 Fn 𝐵) |
10 | fnopfvb 6961 | . . . . . . 7 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐶 ∈ 𝐵) → ((◡𝐹‘𝐶) = 𝑥 ↔ 〈𝐶, 𝑥〉 ∈ ◡𝐹)) | |
11 | 10 | 3adant3 1131 | . . . . . 6 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝐶) = 𝑥 ↔ 〈𝐶, 𝑥〉 ∈ ◡𝐹)) |
12 | 9, 11 | syl3an1 1162 | . . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝐶) = 𝑥 ↔ 〈𝐶, 𝑥〉 ∈ ◡𝐹)) |
13 | 7, 12 | bitrd 279 | . . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ 〈𝐶, 𝑥〉 ∈ ◡𝐹)) |
14 | 13 | 3expa 1117 | . . 3 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ 〈𝐶, 𝑥〉 ∈ ◡𝐹)) |
15 | 14 | reubidva 3394 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶 ↔ ∃!𝑥 ∈ 𝐴 〈𝐶, 𝑥〉 ∈ ◡𝐹)) |
16 | 5, 15 | mpbird 257 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∃!wreu 3376 〈cop 4637 ◡ccnv 5688 Fn wfn 6558 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 |
This theorem is referenced by: 1arith2 16962 uspgredgiedg 29207 disjrdx 32611 ply1divalg3 35627 reuf1odnf 47057 reuf1od 47058 |
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