Theorem oppffn 49621

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 49620	. 2 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
2		opex 5410	. . 3 ⊢ ⟨𝑓, tpos 𝑔⟩ ∈ V
3		0ex 5236	. . 3 ⊢ ∅ ∈ V
4	2, 3	ifex 4512	. 2 ⊢ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅) ∈ V
5	1, 4	fnmpoi 8019	1 ⊢ oppFunc Fn (V × V)

Syntax hints:   ∧ wa 396  Vcvv 3432  ∅c0 4268  ifcif 4461  ⟨cop 4568   × cxp 5623  dom cdm 5625  Rel wrel 5630   Fn wfn 6487  tpos ctpos 8172   oppFunc coppf 49619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-oppf 49620

This theorem is referenced by:  oppfrcl  49625  eloppf  49630  oppff1  49645