Theorem oppff1o 49734

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 49733	. . 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 6756	. . . 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 6863	. . . . . 6 ⊢ (𝑔 = ( oppFunc ‘𝑓) → (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑔) = (( oppFunc ↾ (𝐶 Func 𝐷))‘( oppFunc ‘𝑓)))
8	7	eqeq2d 2772	. . . . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → 𝐶 ∈ 𝑉)
11		oppff1o.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
12	11	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → 𝐷 ∈ 𝑊)
13		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → 𝑓 ∈ (𝑂 Func 𝑃))
14	1, 2, 10, 12, 13	2oppffunc 49731	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → ( oppFunc ‘𝑓) ∈ (𝐶 Func 𝐷))
15	14	fvresd 6883	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → (( oppFunc ↾ (𝐶 Func 𝐷))‘( oppFunc ‘𝑓)) = ( oppFunc ‘( oppFunc ‘𝑓)))
16		relfunc 17878	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
17		eqid 2761	. . . . . . 7 ⊢ ( oppFunc ‘𝑓) = ( oppFunc ‘𝑓)
18	14, 16, 17	2oppf 49717	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → ( oppFunc ‘( oppFunc ‘𝑓)) = 𝑓)
19	15, 18	eqtr2d 2797	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → 𝑓 = (( oppFunc ↾ (𝐶 Func 𝐷))‘( oppFunc ‘𝑓)))
20	8, 14, 19	rspcedvdw 3584	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → ∃𝑔 ∈ (𝐶 Func 𝐷)𝑓 = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑔))
21	20	ralrimiva 3153	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ (𝑂 Func 𝑃)∃𝑔 ∈ (𝐶 Func 𝐷)𝑓 = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑔))
22		dffo3 7079	. . 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 592	. 2 ⊢ (𝜑 → ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–onto→(𝑂 Func 𝑃))
24		df-f1o 6524	. 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 592	1 ⊢ (𝜑 → ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085   ↾ cres 5647  ⟶wf 6513  –1-1→wf1 6514  –onto→wfo 6515  –1-1-onto→wf1o 6516  ‘cfv 6517  (class class class)co 7392  oppCatcoppc 17726   Func cfunc 17870   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-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  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-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  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-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  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-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  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-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-tpos 8201  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-er 8673  df-map 8805  df-ixp 8876  df-en 8924  df-dom 8925  df-sdom 8926  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12479  df-z 12566  df-dec 12686  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-hom 17293  df-cco 17294  df-cat 17683  df-cid 17684  df-homf 17685  df-comf 17686  df-oppc 17727  df-func 17874  df-oppf 49708

This theorem is referenced by:  fucoppc  49995