Theorem rnmptpr 45821

Description: Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
rnmptpr.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
rnmptpr.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
rnmptpr.f	⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
rnmptpr.d	⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
rnmptpr.e	⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)

Assertion

Ref	Expression
rnmptpr	⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐸

Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem rnmptpr

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnmptpr.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		rnmptpr.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		rnmptpr.d	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
4	3	eqeq2d 2780	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐷))
5		rnmptpr.e	. . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
6	5	eqeq2d 2780	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐸))
7	4, 6	rexprg 4668	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)))
8	1, 2, 7	syl2anc 595	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)))
9		rnmptpr.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
10	9	elrnmpt 5949	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶))
11	10	elv 3468	. . 3 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)
12		vex 3467	. . . 4 ⊢ 𝑦 ∈ V
13	12	elpr 4619	. . 3 ⊢ (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))
14	8, 11, 13	3bitr4g 317	. 2 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸}))
15	14	eqrdv 2767	1 ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸})

Syntax hints:   → wi 4   ↔ wb 209   ∨ wo 860   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  Vcvv 3463  {cpr 4596   ↦ cmpt 5196  ran crn 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-mpt 5197  df-cnv 5670  df-dm 5672  df-rn 5673

This theorem is referenced by:  sge0pr  47034