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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 485 | . . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) | |
2 | f1ocnv 6621 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
3 | f1of 6609 | . . . 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 6983 | . . 3 ⊢ ((◡𝐹:𝐵⟶𝐴 ∧ 𝐵 ∈ 𝑉) → ◡𝐹 ∈ V) | |
6 | 4, 5 | sylancom 590 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ 𝑉) → ◡𝐹 ∈ V) |
7 | f1orel 6612 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹) | |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ 𝑉) → Rel 𝐹) |
9 | relcnvexb 7625 | . . 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 259 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 Vcvv 3494 ◡ccnv 5548 Rel wrel 5554 ⟶wf 6345 –1-1-onto→wf1o 6348 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 |
This theorem is referenced by: gsumzf1o 19026 poimirlem3 34889 poimirlem24 34910 poimirlem25 34911 |
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