diag1f1 - Mathbox for Zhi Wang

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Theorem diag1f1 49802

Description: The object part of the diagonal functor is 1-1 if 𝐵 is non-empty. (Contributed by Zhi Wang, 19-Oct-2025.)

**Hypotheses**
Ref	Expression
diag1f1.l	⊢ 𝐿 = (𝐶Δ_func𝐷)
diag1f1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
diag1f1.d	⊢ (𝜑 → 𝐷 ∈ Cat)
diag1f1.a	⊢ 𝐴 = (Base‘𝐶)
diag1f1.b	⊢ 𝐵 = (Base‘𝐷)
diag1f1.0	⊢ (𝜑 → 𝐵 ≠ ∅)

**Assertion**
Ref	Expression
diag1f1	⊢ (𝜑 → (1^st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶))

Proof of Theorem diag1f1 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		diag1f1.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
2		eqid 2737	. . . 4 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
3	2	fucbas 17927	. . 3 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
4		diag1f1.l	. . . . 5 ⊢ 𝐿 = (𝐶Δ_func𝐷)
5		diag1f1.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		diag1f1.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
7	4, 5, 6, 2	diagcl 18204	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
8	7	func1st2nd 49571	. . 3 ⊢ (𝜑 → (1^st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2^nd ‘𝐿))
9	1, 3, 8	funcf1 17830	. 2 ⊢ (𝜑 → (1^st ‘𝐿):𝐴⟶(𝐷 Func 𝐶))
10	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐶 ∈ Cat)
11	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐷 ∈ Cat)
12		diag1f1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
13		diag1f1.0	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ ∅)
14	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐵 ≠ ∅)
15		simprl 771	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
16		simprr 773	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
17		eqid 2737	. . . 4 ⊢ ((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑥)
18		eqid 2737	. . . 4 ⊢ ((1^st ‘𝐿)‘𝑦) = ((1^st ‘𝐿)‘𝑦)
19	4, 10, 11, 1, 12, 14, 15, 16, 17, 18	diag1f1lem 49801	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑦) → 𝑥 = 𝑦))
20	19	ralrimivva 3181	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑦) → 𝑥 = 𝑦))
21		dff13 7206	. 2 ⊢ ((1^st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶) ↔ ((1^st ‘𝐿):𝐴⟶(𝐷 Func 𝐶) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑦) → 𝑥 = 𝑦)))
22	9, 20, 21	sylanbrc 584	1 ⊢ (𝜑 → (1^st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∅c0 4274 ⟶wf 6492 –1-1→wf1 6493 ‘cfv 6496 (class class class)co 7364 1^st c1st 7937 2^nd c2nd 7938 Basecbs 17176 Catccat 17627 Func cfunc 17818 FuncCat cfuc 17909 Δ_funccdiag 18175

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-mulrcl 11098 ax-mulcom 11099 ax-addass 11100 ax-mulass 11101 ax-distr 11102 ax-i2m1 11103 ax-1ne0 11104 ax-1rid 11105 ax-rnegex 11106 ax-rrecex 11107 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110 ax-pre-ltadd 11111 ax-pre-mulgt0 11112

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5523 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5581 df-we 5583 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7321 df-ov 7367 df-oprab 7368 df-mpo 7369 df-om 7815 df-1st 7939 df-2nd 7940 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-sub 11376 df-neg 11377 df-nn 12172 df-2 12241 df-3 12242 df-4 12243 df-5 12244 df-6 12245 df-7 12246 df-8 12247 df-9 12248 df-n0 12435 df-z 12522 df-dec 12642 df-uz 12786 df-fz 13459 df-struct 17114 df-slot 17149 df-ndx 17161 df-base 17177 df-hom 17241 df-cco 17242 df-cat 17631 df-cid 17632 df-func 17822 df-nat 17910 df-fuc 17911 df-xpc 18135 df-1stf 18136 df-curf 18177 df-diag 18179

This theorem is referenced by: eufunclem 50016 diag1f1o 50029

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