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Mirrors > Home > MPE Home > Th. List > eldmrexrn | Structured version Visualization version GIF version |
Description: For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.) |
Ref | Expression |
---|---|
eldmrexrn | ⊢ (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvelrn 6709 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ dom 𝐹) → (𝐹‘𝑌) ∈ ran 𝐹) | |
2 | eqid 2795 | . . 3 ⊢ (𝐹‘𝑌) = (𝐹‘𝑌) | |
3 | eqeq1 2799 | . . . 4 ⊢ (𝑥 = (𝐹‘𝑌) → (𝑥 = (𝐹‘𝑌) ↔ (𝐹‘𝑌) = (𝐹‘𝑌))) | |
4 | 3 | rspcev 3559 | . . 3 ⊢ (((𝐹‘𝑌) ∈ ran 𝐹 ∧ (𝐹‘𝑌) = (𝐹‘𝑌)) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)) |
5 | 1, 2, 4 | sylancl 586 | . 2 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ dom 𝐹) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)) |
6 | 5 | ex 413 | 1 ⊢ (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∃wrex 3106 dom cdm 5443 ran crn 5444 Fun wfun 6219 ‘cfv 6225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-iota 6189 df-fun 6227 df-fn 6228 df-fv 6233 |
This theorem is referenced by: eldmrexrnb 6723 |
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