Theorem elunirnALT 7181

Description: Alternate proof of elunirn 7180. It is shorter but requires ax-pow 5301 (through eluniima 7179, funiunfv 7177, ndmfv 6849). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elunirnALT	⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem elunirnALT

Step	Hyp	Ref	Expression
1		imadmrn 6016	. . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2	1	unieqi 4869	. . 3 ⊢ ∪ (𝐹 “ dom 𝐹) = ∪ ran 𝐹
3	2	eleq2i 2821	. 2 ⊢ (𝐴 ∈ ∪ (𝐹 “ dom 𝐹) ↔ 𝐴 ∈ ∪ ran 𝐹)
4		eluniima 7179	. 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)))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2110  ∃wrex 3054  ∪ cuni 4857  dom cdm 5614  ran crn 5615   “ cima 5617  Fun wfun 6471  ‘cfv 6477

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 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-fv 6485

This theorem is referenced by: (None)