Theorem oppffn 49041

Description: oppFunc is a function on (V × V). (Contributed by Zhi Wang, 17-Nov-2025.)

Assertion

Ref	Expression
oppffn	⊢ oppFunc Fn (V × V)

Proof of Theorem oppffn

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

Step	Hyp	Ref	Expression
1		df-oppf 49040	. 2 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
2		opex 5432	. . 3 ⊢ ⟨𝑓, tpos 𝑔⟩ ∈ V
3		0ex 5270	. . 3 ⊢ ∅ ∈ V
4	2, 3	ifex 4547	. 2 ⊢ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅) ∈ V
5	1, 4	fnmpoi 8058	1 ⊢ oppFunc Fn (V × V)

Syntax hints:   ∧ wa 395  Vcvv 3455  ∅c0 4304  ifcif 4496  ⟨cop 4603   × cxp 5644  dom cdm 5646  Rel wrel 5651   Fn wfn 6514  tpos ctpos 8213  oppFunccoppf 49039

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 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395  ax-un 7718

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-fv 6527  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-oppf 49040

This theorem is referenced by:  oppff1  49060