Theorem elunirn2OLD 7258

Description: Obsolete version of elfvunirn 6923 as of 12-Jan-2025. (Contributed by Thierry Arnoux, 13-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
elunirn2OLD	⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → 𝐵 ∈ ∪ ran 𝐹)

Proof of Theorem elunirn2OLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6928	. . . 4 ⊢ (𝐵 ∈ (𝐹‘𝐴) → 𝐴 ∈ dom 𝐹)
2		fveq2 6891	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
3	2	eleq2d 2811	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ (𝐹‘𝑥) ↔ 𝐵 ∈ (𝐹‘𝐴)))
4	3	rspcev 3602	. . . 4 ⊢ ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥))
5	1, 4	mpancom 686	. . 3 ⊢ (𝐵 ∈ (𝐹‘𝐴) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥))
6	5	adantl 480	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥))
7		elunirn 7256	. . 3 ⊢ (Fun 𝐹 → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥)))
8	7	adantr 479	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹‘𝑥)))
9	6, 8	mpbird 256	1 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (𝐹‘𝐴)) → 𝐵 ∈ ∪ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3060  ∪ cuni 4903  dom cdm 5672  ran crn 5673  Fun wfun 6536  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550

This theorem is referenced by: (None)