Theorem rnmptpr 45168

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 2747	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐷))
5		rnmptpr.e	. . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
6	5	eqeq2d 2747	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐸))
7	4, 6	rexprg 4678	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)))
8	1, 2, 7	syl2anc 584	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)))
9		rnmptpr.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
10	9	elrnmpt 5943	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶))
11	10	elv 3469	. . 3 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)
12		vex 3468	. . . 4 ⊢ 𝑦 ∈ V
13	12	elpr 4631	. . 3 ⊢ (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))
14	8, 11, 13	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸}))
15	14	eqrdv 2734	1 ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸})

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  Vcvv 3464  {cpr 4608   ↦ cmpt 5206  ran crn 5660

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-mpt 5207  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is referenced by:  sge0pr  46390