Theorem oppfvalg 49617

Description: Value of the opposite functor. (Contributed by Zhi Wang, 13-Nov-2025.)

Assertion

Ref	Expression
oppfvalg	⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 oppFunc 𝐺) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))

Proof of Theorem oppfvalg

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
2	1	releqd 5730	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (Rel 𝑔 ↔ Rel 𝐺))
3	1	dmeqd 5856	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → dom 𝑔 = dom 𝐺)
4	3	releqd 5730	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (Rel dom 𝑔 ↔ Rel dom 𝐺))
5	2, 4	anbi12d 633	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((Rel 𝑔 ∧ Rel dom 𝑔) ↔ (Rel 𝐺 ∧ Rel dom 𝐺)))
6		simpl 482	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
7	1	tposeqd 8174	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → tpos 𝑔 = tpos 𝐺)
8	6, 7	opeq12d 4825	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ⟨𝑓, tpos 𝑔⟩ = ⟨𝐹, tpos 𝐺⟩)
9	5, 8	ifbieq1d 4492	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))
10		df-oppf 49614	. 2 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
11		opex 5413	. . 3 ⊢ ⟨𝐹, tpos 𝐺⟩ ∈ V
12		0ex 5243	. . 3 ⊢ ∅ ∈ V
13	11, 12	ifex 4518	. 2 ⊢ if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) ∈ V
14	9, 10, 13	ovmpoa 7517	1 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 oppFunc 𝐺) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ∅c0 4274  ifcif 4467  ⟨cop 4574  dom cdm 5626  Rel wrel 5631  (class class class)co 7362  tpos ctpos 8170   oppFunc coppf 49613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-res 5638  df-iota 6450  df-fun 6496  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-tpos 8171  df-oppf 49614

This theorem is referenced by:  oppfrcl3  49621  oppf1st2nd  49622  2oppf  49623  eloppf  49624  eloppf2  49625  oppfval  49627