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Theorem fmptpr 6602
Description: Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)
Hypotheses
Ref Expression
fmptpr.1 (𝜑𝐴𝑉)
fmptpr.2 (𝜑𝐵𝑊)
fmptpr.3 (𝜑𝐶𝑋)
fmptpr.4 (𝜑𝐷𝑌)
fmptpr.5 ((𝜑𝑥 = 𝐴) → 𝐸 = 𝐶)
fmptpr.6 ((𝜑𝑥 = 𝐵) → 𝐸 = 𝐷)
Assertion
Ref Expression
fmptpr (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝜑,𝑥
Allowed substitution hints:   𝐸(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)   𝑌(𝑥)

Proof of Theorem fmptpr
StepHypRef Expression
1 df-pr 4324 . . 3 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
21a1i 11 . 2 (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}))
3 fmptpr.5 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐸 = 𝐶)
4 fmptpr.1 . . . 4 (𝜑𝐴𝑉)
5 fmptpr.3 . . . 4 (𝜑𝐶𝑋)
63, 4, 5fmptsnd 6599 . . 3 (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐸))
76uneq1d 3909 . 2 (𝜑 → ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = ((𝑥 ∈ {𝐴} ↦ 𝐸) ∪ {⟨𝐵, 𝐷⟩}))
8 fmptpr.2 . . . 4 (𝜑𝐵𝑊)
9 elex 3352 . . . 4 (𝐵𝑊𝐵 ∈ V)
108, 9syl 17 . . 3 (𝜑𝐵 ∈ V)
11 fmptpr.4 . . . 4 (𝜑𝐷𝑌)
12 elex 3352 . . . 4 (𝐷𝑌𝐷 ∈ V)
1311, 12syl 17 . . 3 (𝜑𝐷 ∈ V)
14 df-pr 4324 . . . . 5 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
1514eqcomi 2769 . . . 4 ({𝐴} ∪ {𝐵}) = {𝐴, 𝐵}
1615a1i 11 . . 3 (𝜑 → ({𝐴} ∪ {𝐵}) = {𝐴, 𝐵})
17 fmptpr.6 . . 3 ((𝜑𝑥 = 𝐵) → 𝐸 = 𝐷)
1810, 13, 16, 17fmptapd 6601 . 2 (𝜑 → ((𝑥 ∈ {𝐴} ↦ 𝐸) ∪ {⟨𝐵, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸))
192, 7, 183eqtrd 2798 1 (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cun 3713  {csn 4321  {cpr 4323  cop 4327  cmpt 4881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-mpt 4882
This theorem is referenced by:  pmtrprfvalrn  18108  esumsnf  30435  sge0sn  41099  zlmodzxzscm  42645  zlmodzxzadd  42646
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