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Mirrors > Home > MPE Home > Th. List > elunirnALT | Structured version Visualization version GIF version |
Description: Alternate proof of elunirn 7194. It is shorter but requires ax-pow 5318 (through eluniima 7193, funiunfv 7191, ndmfv 6874). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elunirnALT | ⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imadmrn 6021 | . . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
2 | 1 | unieqi 4876 | . . 3 ⊢ ∪ (𝐹 “ dom 𝐹) = ∪ ran 𝐹 |
3 | 2 | eleq2i 2829 | . 2 ⊢ (𝐴 ∈ ∪ (𝐹 “ dom 𝐹) ↔ 𝐴 ∈ ∪ ran 𝐹) |
4 | eluniima 7193 | . 2 ⊢ (Fun 𝐹 → (𝐴 ∈ ∪ (𝐹 “ dom 𝐹) ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) | |
5 | 3, 4 | bitr3id 284 | 1 ⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 ∃wrex 3071 ∪ cuni 4863 dom cdm 5631 ran crn 5632 “ cima 5634 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-iun 4954 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-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-fv 6501 |
This theorem is referenced by: (None) |
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