Theorem rnmptpr 39358

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		vex 3203	. . . . . 6 ⊢ 𝑦 ∈ V
2		rnmptpr.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)
3	2	elrnmpt 5372	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶))
4	1, 3	ax-mp 5	. . . . 5 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)
5	4	a1i 11	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶))
6		rnmptpr.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		rnmptpr.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
8		rnmptpr.d	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
9	8	eqeq2d 2632	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐷))
10		rnmptpr.e	. . . . . . 7 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
11	10	eqeq2d 2632	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐸))
12	9, 11	rexprg 4235	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)))
13	6, 7, 12	syl2anc 693	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)))
14	1	elpr 4198	. . . . . 6 ⊢ (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))
15	14	bicomi 214	. . . . 5 ⊢ ((𝑦 = 𝐷 ∨ 𝑦 = 𝐸) ↔ 𝑦 ∈ {𝐷, 𝐸})
16	15	a1i 11	. . . 4 ⊢ (𝜑 → ((𝑦 = 𝐷 ∨ 𝑦 = 𝐸) ↔ 𝑦 ∈ {𝐷, 𝐸}))
17	5, 13, 16	3bitrd 294	. . 3 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸}))
18	17	alrimiv 1855	. 2 ⊢ (𝜑 → ∀𝑦(𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸}))
19		dfcleq 2616	. 2 ⊢ (ran 𝐹 = {𝐷, 𝐸} ↔ ∀𝑦(𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸}))
20	18, 19	sylibr 224	1 ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸})

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200  {cpr 4179   ↦ cmpt 4729  ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  sge0pr  40611