Theorem oppff1o 49646

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 49645	. . 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 2751	. . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → 𝐶 ∈ 𝑉)
11		oppff1o.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
12	11	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → 𝐷 ∈ 𝑊)
13		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → 𝑓 ∈ (𝑂 Func 𝑃))
14	1, 2, 10, 12, 13	2oppffunc 49643	. . . . 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 17827	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
17		eqid 2740	. . . . . . 7 ⊢ ( oppFunc ‘𝑓) = ( oppFunc ‘𝑓)
18	14, 16, 17	2oppf 49629	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → ( oppFunc ‘( oppFunc ‘𝑓)) = 𝑓)
19	15, 18	eqtr2d 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → 𝑓 = (( oppFunc ↾ (𝐶 Func 𝐷))‘( oppFunc ‘𝑓)))
20	8, 14, 19	rspcedvdw 3570	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑂 Func 𝑃)) → ∃𝑔 ∈ (𝐶 Func 𝐷)𝑓 = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑔))
21	20	ralrimiva 3132	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ (𝑂 Func 𝑃)∃𝑔 ∈ (𝐶 Func 𝐷)𝑓 = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑔))
22		dffo3 7050	. . 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 589	. 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 589	1 ⊢ (𝜑 → ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064   ↾ cres 5627  ⟶wf 6488  –1-1→wf1 6489  –onto→wfo 6490  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7363  oppCatcoppc 17675   Func cfunc 17819   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-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  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-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  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-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  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-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  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-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 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-tpos 8173  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-map 8772  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-n0 12436  df-z 12523  df-dec 12643  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-hom 17242  df-cco 17243  df-cat 17632  df-cid 17633  df-homf 17634  df-comf 17635  df-oppc 17676  df-func 17823  df-oppf 49620

This theorem is referenced by:  fucoppc  49907