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Mirrors > Home > MPE Home > Th. List > elunirnALT | Structured version Visualization version GIF version |
Description: Alternate proof of elunirn 7004. It is shorter but requires ax-pow 5259 (through eluniima 7003, funiunfv 7001, ndmfv 6695). (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 5934 | . . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
2 | 1 | unieqi 4841 | . . 3 ⊢ ∪ (𝐹 “ dom 𝐹) = ∪ ran 𝐹 |
3 | 2 | eleq2i 2904 | . 2 ⊢ (𝐴 ∈ ∪ (𝐹 “ dom 𝐹) ↔ 𝐴 ∈ ∪ ran 𝐹) |
4 | eluniima 7003 | . 2 ⊢ (Fun 𝐹 → (𝐴 ∈ ∪ (𝐹 “ dom 𝐹) ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) | |
5 | 3, 4 | syl5bbr 287 | 1 ⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2110 ∃wrex 3139 ∪ cuni 4832 dom cdm 5550 ran crn 5551 “ cima 5553 Fun wfun 6344 ‘cfv 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-fv 6358 |
This theorem is referenced by: (None) |
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