Theorem oppffn 49709

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 49708	. 2 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
2		opex 5430	. . 3 ⊢ ⟨𝑓, tpos 𝑔⟩ ∈ V
3		0ex 5256	. . 3 ⊢ ∅ ∈ V
4	2, 3	ifex 4530	. 2 ⊢ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅) ∈ V
5	1, 4	fnmpoi 8047	1 ⊢ oppFunc Fn (V × V)

Syntax hints:   ∧ wa 399  Vcvv 3453  ∅c0 4285  ifcif 4479  ⟨cop 4587   × cxp 5643  dom cdm 5645  Rel wrel 5650   Fn wfn 6512  tpos ctpos 8200   oppFunc coppf 49707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-oppf 49708

This theorem is referenced by:  oppfrcl  49713  eloppf  49718  oppff1  49733