Theorem oppffn 49754

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 49753	. 2 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
2		opex 5435	. . 3 ⊢ ⟨𝑓, tpos 𝑔⟩ ∈ V
3		0ex 5261	. . 3 ⊢ ∅ ∈ V
4	2, 3	ifex 4534	. 2 ⊢ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅) ∈ V
5	1, 4	fnmpoi 8055	1 ⊢ oppFunc Fn (V × V)

Syntax hints:   ∧ wa 400  Vcvv 3457  ∅c0 4288  ifcif 4483  ⟨cop 4591   × cxp 5649  dom cdm 5651  Rel wrel 5656   Fn wfn 6520  tpos ctpos 8209   oppFunc coppf 49752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-oppf 49753

This theorem is referenced by:  oppfrcl  49758  eloppf  49763  oppff1  49778