Theorem elunirnALT 7192

Description: Alternate proof of elunirn 7191. It is shorter but requires ax-pow 5305 (through eluniima 7190, funiunfv 7188, ndmfv 6860). (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 6024	. . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2	1	unieqi 4870	. . 3 ⊢ ∪ (𝐹 “ dom 𝐹) = ∪ ran 𝐹
3	2	eleq2i 2823	. 2 ⊢ (𝐴 ∈ ∪ (𝐹 “ dom 𝐹) ↔ 𝐴 ∈ ∪ ran 𝐹)
4		eluniima 7190	. 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 2111  ∃wrex 3056  ∪ cuni 4858  dom cdm 5619  ran crn 5620   “ cima 5622  Fun wfun 6481  ‘cfv 6487

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

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 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-fv 6495

This theorem is referenced by: (None)