diag1f1 - Mathbox for Zhi Wang

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Theorem diag1f1 49185

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 2736	. . . 4 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
3	2	fucbas 17981	. . 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 18258	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
8	7	func1st2nd 49010	. . 3 ⊢ (𝜑 → (1^st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2^nd ‘𝐿))
9	1, 3, 8	funcf1 17884	. 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 770	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
16		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
17		eqid 2736	. . . 4 ⊢ ((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑥)
18		eqid 2736	. . . 4 ⊢ ((1^st ‘𝐿)‘𝑦) = ((1^st ‘𝐿)‘𝑦)
19	4, 10, 11, 1, 12, 14, 15, 16, 17, 18	diag1f1lem 49184	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑦) → 𝑥 = 𝑦))
20	19	ralrimivva 3188	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑦) → 𝑥 = 𝑦))
21		dff13 7252	. 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 583	1 ⊢ (𝜑 → (1^st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 ∅c0 4313 ⟶wf 6532 –1-1→wf1 6533 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 Basecbs 17233 Catccat 17681 Func cfunc 17872 FuncCat cfuc 17963 Δ_funccdiag 18229

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-fz 13530 df-struct 17171 df-slot 17206 df-ndx 17218 df-base 17234 df-hom 17300 df-cco 17301 df-cat 17685 df-cid 17686 df-func 17876 df-nat 17964 df-fuc 17965 df-xpc 18189 df-1stf 18190 df-curf 18231 df-diag 18233

This theorem is referenced by: eufunclem 49373 diag1f1o 49386

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