Theorem elrnmpog 7279

Description: Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
rngop.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
elrnmpog	⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐷

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem elrnmpog

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2824	. . 3 ⊢ (𝑧 = 𝐷 → (𝑧 = 𝐶 ↔ 𝐷 = 𝐶))
2	1	2rexbidv 3299	. 2 ⊢ (𝑧 = 𝐷 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶))
3		rngop.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	rnmpo 7277	. 2 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
5	2, 4	elab2g 3664	1 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1536   ∈ wcel 2113  ∃wrex 3138  ran crn 5549   ∈ cmpo 7151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-br 5060  df-opab 5122  df-cnv 5556  df-dm 5558  df-rn 5559  df-oprab 7153  df-mpo 7154

This theorem is referenced by:  ordtbas2  21794  txopn  22205  tgisline  26411  elsx  31474  smflimlem6  43126