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Mirrors > Home > MPE Home > Th. List > Mathboxes > elunirn2 | Structured version Visualization version GIF version |
Description: Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.) |
Ref | Expression |
---|---|
elunirn2 | ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → 𝐵 ∈ ∪ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6130 | . . . 4 ⊢ (𝐵 ∈ (𝐹‘𝐴) → 𝐴 ∈ dom 𝐹) | |
2 | fveq2 6103 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
3 | 2 | eleq2d 2673 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ (𝐹‘𝑥) ↔ 𝐵 ∈ (𝐹‘𝐴))) |
4 | 3 | rspcev 3282 | . . . 4 ⊢ ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥)) |
5 | 1, 4 | mpancom 700 | . . 3 ⊢ (𝐵 ∈ (𝐹‘𝐴) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥)) |
6 | 5 | adantl 481 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥)) |
7 | elunirn 6413 | . . 3 ⊢ (Fun 𝐹 → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥))) | |
8 | 7 | adantr 480 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥))) |
9 | 6, 8 | mpbird 246 | 1 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → 𝐵 ∈ ∪ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∪ cuni 4372 dom cdm 5038 ran crn 5039 Fun wfun 5798 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: measbasedom 29592 sxbrsigalem0 29660 |
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