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Mirrors > Home > MPE Home > Th. List > foelrn | Structured version Visualization version GIF version |
Description: Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.) |
Ref | Expression |
---|---|
foelrn | ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffo3 7093 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) | |
2 | 1 | simprbi 496 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
3 | eqeq1 2728 | . . . 4 ⊢ (𝑦 = 𝐶 → (𝑦 = (𝐹‘𝑥) ↔ 𝐶 = (𝐹‘𝑥))) | |
4 | 3 | rexbidv 3170 | . . 3 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥))) |
5 | 4 | rspccva 3603 | . 2 ⊢ ((∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥)) |
6 | 2, 5 | sylan 579 | 1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 ⟶wf 6529 –onto→wfo 6531 ‘cfv 6533 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fo 6539 df-fv 6541 |
This theorem is referenced by: foco2 7100 fofinf1o 9322 fodomacn 10046 iunfictbso 10104 cff1 10248 cofsmo 10259 axcclem 10447 konigthlem 10558 tskuni 10773 fulli 17862 efgredlemc 19650 efgrelexlemb 19655 efgredeu 19657 ghmcyg 19801 znfld 21416 znrrg 21421 cygznlem3 21425 ovoliunnul 25346 lgsdchr 27192 foresf1o 32166 iunrdx 32219 znfermltl 32910 crngohomfo 37330 fourierdlem20 45294 fourierdlem52 45325 fourierdlem63 45336 fourierdlem64 45337 fourierdlem65 45338 isomuspgrlem2d 46950 |
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