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Mirrors > Home > MPE Home > Th. List > elunirnALT | Structured version Visualization version GIF version |
Description: Alternate proof of elunirn 7199. It is shorter but requires ax-pow 5321 (through eluniima 7198, funiunfv 7196, ndmfv 6878). (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 6024 | . . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
2 | 1 | unieqi 4879 | . . 3 ⊢ ∪ (𝐹 “ dom 𝐹) = ∪ ran 𝐹 |
3 | 2 | eleq2i 2826 | . 2 ⊢ (𝐴 ∈ ∪ (𝐹 “ dom 𝐹) ↔ 𝐴 ∈ ∪ ran 𝐹) |
4 | eluniima 7198 | . 2 ⊢ (Fun 𝐹 → (𝐴 ∈ ∪ (𝐹 “ dom 𝐹) ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) | |
5 | 3, 4 | bitr3id 285 | 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 2107 ∃wrex 3070 ∪ cuni 4866 dom cdm 5634 ran crn 5635 “ cima 5637 Fun wfun 6491 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 |
This theorem is referenced by: (None) |
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