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Mirrors > Home > MPE Home > Th. List > foelrnf | Structured version Visualization version GIF version |
Description: Property of a surjective function. As foelrn 7113 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
foelrnf.1 | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
foelrnf | ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | foelrnf.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
2 | 1 | dffo3f 7112 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) |
3 | 2 | simprbi 495 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
4 | eqeq1 2730 | . . . 4 ⊢ (𝑦 = 𝐶 → (𝑦 = (𝐹‘𝑥) ↔ 𝐶 = (𝐹‘𝑥))) | |
5 | 4 | rexbidv 3169 | . . 3 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥))) |
6 | 5 | rspccva 3606 | . 2 ⊢ ((∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥)) |
7 | 3, 6 | sylan 578 | 1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Ⅎwnfc 2876 ∀wral 3051 ∃wrex 3060 ⟶wf 6542 –onto→wfo 6544 ‘cfv 6546 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-fo 6552 df-fv 6554 |
This theorem is referenced by: fompt 7124 |
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