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Theorem elmapex 8429

Description: Eliminate antecedent for mapping theorems: domain can be taken to be a set. (Contributed by Stefan O'Rear, 8-Oct-2014.)

**Assertion**
Ref	Expression
elmapex	⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V))

**Proof of Theorem elmapex**
Step	Hyp	Ref	Expression
1		n0i 4301	. 2 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → ¬ (𝐵 ↑_m 𝐶) = ∅)
2		fnmap 8415	. . . 4 ⊢ ↑_m Fn (V × V)
3		fndm 6457	. . . 4 ⊢ ( ↑_m Fn (V × V) → dom ↑_m = (V × V))
4	2, 3	ax-mp 5	. . 3 ⊢ dom ↑_m = (V × V)
5	4	ndmov 7334	. 2 ⊢ (¬ (𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵 ↑_m 𝐶) = ∅)
6	1, 5	nsyl2 143	1 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 × cxp 5555 dom cdm 5557 Fn wfn 6352 (class class class)co 7158 ↑_m cmap 8408

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410

This theorem is referenced by: elmapi 8430 elmapssres 8433 mapsspm 8442 elmapresaun 8446 mapss 8455 ralxpmap 8462 mapdom1 8684 wemapwe 9162 isf34lem6 9804 mndvcl 21004 mndvass 21005 mndvlid 21006 mndvrid 21007 grpvlinv 21008 grpvrinv 21009 mhmvlin 21010 tposmap 21068 satfv1lem 32611 mapfzcons 39320 ovnhoilem2 42891

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