MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fmptap Structured version   Visualization version   GIF version

Theorem fmptap 6314
Description: Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
fmptap.0a 𝐴 ∈ V
fmptap.0b 𝐵 ∈ V
fmptap.1 (𝑅 ∪ {𝐴}) = 𝑆
fmptap.2 (𝑥 = 𝐴𝐶 = 𝐵)
Assertion
Ref Expression
fmptap ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑆
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem fmptap
StepHypRef Expression
1 fmptap.0a . . . . 5 𝐴 ∈ V
2 fmptap.0b . . . . 5 𝐵 ∈ V
3 fmptsn 6311 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
41, 2, 3mp2an 703 . . . 4 {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵)
5 elsni 4136 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
6 fmptap.2 . . . . . 6 (𝑥 = 𝐴𝐶 = 𝐵)
75, 6syl 17 . . . . 5 (𝑥 ∈ {𝐴} → 𝐶 = 𝐵)
87mpteq2ia 4657 . . . 4 (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵)
94, 8eqtr4i 2629 . . 3 {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶)
109uneq2i 3720 . 2 ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
11 mptun 5919 . 2 (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
12 fmptap.1 . . 3 (𝑅 ∪ {𝐴}) = 𝑆
13 mpteq1 4654 . . 3 ((𝑅 ∪ {𝐴}) = 𝑆 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥𝑆𝐶))
1412, 13ax-mp 5 . 2 (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥𝑆𝐶)
1510, 11, 143eqtr2i 2632 1 ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  Vcvv 3167  cun 3532  {csn 4119  cop 4125  cmpt 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator