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Mirrors > Home > MPE Home > Th. List > f1oexrnex | Structured version Visualization version GIF version |
Description: If the range of a 1-1 onto function is a set, the function itself is a set. (Contributed by AV, 2-Jun-2019.) |
Ref | Expression |
---|---|
f1oexrnex | ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) | |
2 | f1ocnv 6846 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
3 | f1of 6834 | . . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴) | |
4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ 𝑉) → ◡𝐹:𝐵⟶𝐴) |
5 | fex 7234 | . . 3 ⊢ ((◡𝐹:𝐵⟶𝐴 ∧ 𝐵 ∈ 𝑉) → ◡𝐹 ∈ V) | |
6 | 4, 5 | sylancom 586 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ 𝑉) → ◡𝐹 ∈ V) |
7 | f1orel 6837 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹) | |
8 | 7 | adantr 479 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ 𝑉) → Rel 𝐹) |
9 | relcnvexb 7932 | . . 3 ⊢ (Rel 𝐹 → (𝐹 ∈ V ↔ ◡𝐹 ∈ V)) | |
10 | 8, 9 | syl 17 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐹 ∈ V ↔ ◡𝐹 ∈ V)) |
11 | 6, 10 | mpbird 256 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 Vcvv 3463 ◡ccnv 5671 Rel wrel 5677 ⟶wf 6539 –1-1-onto→wf1o 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 |
This theorem is referenced by: gsumzf1o 19871 poimirlem3 37153 poimirlem24 37174 poimirlem25 37175 |
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