Theorem oppff1o 49390

Description: The operation generating opposite functors is bijective. (Contributed by Zhi Wang, 17-Nov-2025.)

Hypotheses

Ref	Expression
oppff1.o	⊢ 𝑂 = (oppCat‘𝐶)
oppff1.p	⊢ 𝑃 = (oppCat‘𝐷)
oppff1o.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
oppff1o.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
oppff1o	⊢ (𝜑 → ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃))

Proof of Theorem oppff1o

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

Step	Hyp	Ref	Expression
1		oppff1.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐶)
2		oppff1.p	. . . 4 ⊢ 𝑃 = (oppCat‘𝐷)
3	1, 2	oppff1 49389	. . 3 ⊢ ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1→(𝑂 Func 𝑃)
4	3	a1i 11	. 2 ⊢ (𝜑 → ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1→(𝑂 Func 𝑃))
5		f1f 6730	. . . 4 ⊢ (( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1→(𝑂 Func 𝑃) → ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)⟶(𝑂 Func 𝑃))
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)⟶(𝑂 Func 𝑃))
7		fveq2 6834	. . . . . 6 ⊢ (𝑔 = ( oppFunc ‘𝑓) → (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑔) = (( oppFunc ↾ (𝐶 Func 𝐷))‘( oppFunc ‘𝑓)))
8	7	eqeq2d 2747	. . . . 5 ⊢ (𝑔 = ( oppFunc ‘𝑓) → (𝑓 = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑔) ↔ 𝑓 = (( oppFunc ↾ (𝐶 Func 𝐷))‘( oppFunc ‘𝑓))))
9		oppff1o.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
10	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → 𝐶 ∈ 𝑉)
11		oppff1o.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → 𝐷 ∈ 𝑊)
13		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → 𝑓 ∈ (𝑂 Func 𝑃))
14	1, 2, 10, 12, 13	2oppffunc 49387	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → ( oppFunc ‘𝑓) ∈ (𝐶 Func 𝐷))
15	14	fvresd 6854	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → (( oppFunc ↾ (𝐶 Func 𝐷))‘( oppFunc ‘𝑓)) = ( oppFunc ‘( oppFunc ‘𝑓)))
16		relfunc 17786	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
17		eqid 2736	. . . . . . 7 ⊢ ( oppFunc ‘𝑓) = ( oppFunc ‘𝑓)
18	14, 16, 17	2oppf 49373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → ( oppFunc ‘( oppFunc ‘𝑓)) = 𝑓)
19	15, 18	eqtr2d 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → 𝑓 = (( oppFunc ↾ (𝐶 Func 𝐷))‘( oppFunc ‘𝑓)))
20	8, 14, 19	rspcedvdw 3579	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → ∃𝑔 ∈ (𝐶 Func 𝐷)𝑓 = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑔))
21	20	ralrimiva 3128	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ (𝑂 Func 𝑃)∃𝑔 ∈ (𝐶 Func 𝐷)𝑓 = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑔))
22		dffo3 7047	. . 3 ⊢ (( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–onto→(𝑂 Func 𝑃) ↔ (( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)⟶(𝑂 Func 𝑃) ∧ ∀𝑓 ∈ (𝑂 Func 𝑃)∃𝑔 ∈ (𝐶 Func 𝐷)𝑓 = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑔)))
23	6, 21, 22	sylanbrc 583	. 2 ⊢ (𝜑 → ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–onto→(𝑂 Func 𝑃))
24		df-f1o 6499	. 2 ⊢ (( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃) ↔ (( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1→(𝑂 Func 𝑃) ∧ ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–onto→(𝑂 Func 𝑃)))
25	4, 23, 24	sylanbrc 583	1 ⊢ (𝜑 → ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060   ↾ cres 5626  ⟶wf 6488  –1-1→wf1 6489  –onto→wfo 6490  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358  oppCatcoppc 17634   Func cfunc 17778   oppFunc coppf 49363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-hom 17201  df-cco 17202  df-cat 17591  df-cid 17592  df-homf 17593  df-comf 17594  df-oppc 17635  df-func 17782  df-oppf 49364

This theorem is referenced by:  fucoppc  49651