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Mirrors > Home > MPE Home > Th. List > elunirn2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of elfvunirn 6871 as of 12-Jan-2025. (Contributed by Thierry Arnoux, 13-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
elunirn2OLD | ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → 𝐵 ∈ ∪ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6876 | . . . 4 ⊢ (𝐵 ∈ (𝐹‘𝐴) → 𝐴 ∈ dom 𝐹) | |
2 | fveq2 6839 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
3 | 2 | eleq2d 2823 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ (𝐹‘𝑥) ↔ 𝐵 ∈ (𝐹‘𝐴))) |
4 | 3 | rspcev 3579 | . . . 4 ⊢ ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥)) |
5 | 1, 4 | mpancom 686 | . . 3 ⊢ (𝐵 ∈ (𝐹‘𝐴) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥)) |
6 | 5 | adantl 482 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥)) |
7 | elunirn 7194 | . . 3 ⊢ (Fun 𝐹 → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥))) | |
8 | 7 | adantr 481 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥))) |
9 | 6, 8 | mpbird 256 | 1 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → 𝐵 ∈ ∪ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3071 ∪ cuni 4863 dom cdm 5631 ran crn 5632 Fun wfun 6487 ‘cfv 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6445 df-fun 6495 df-fn 6496 df-fv 6501 |
This theorem is referenced by: (None) |
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