Theorem oppffn 50201

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 50200	. 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 5261	. . 3 ⊢ ∅ ∈ V
4	2, 3	ifex 4533	. 2 ⊢ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅) ∈ V
5	1, 4	fnmpoi 8079	1 ⊢ oppFunc Fn (V × V)

This proof depends on syntax axioms:   ∧ wa 401  Vcvv 3451  ∅c0 4279  ifcif 4482  ⟨cop 4590   × cxp 5649  dom cdm 5651  Rel wrel 5656   Fn wfn 6532  tpos ctpos 8235   oppFunc coppf 50199

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  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 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-oppf 50200

This theorem is used by:  oppfrcl  50205  eloppf  50210  oppff1  50225