diag1f1 - Mathbox for Zhi Wang

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Theorem diag1f1 49809

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 2741	. . . 4 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
3	2	fucbas 17925	. . 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 18202	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
8	7	func1st2nd 49578	. . 3 ⊢ (𝜑 → (1^st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2^nd ‘𝐿))
9	1, 3, 8	funcf1 17828	. 2 ⊢ (𝜑 → (1^st ‘𝐿):𝐴⟶(𝐷 Func 𝐶))
10	5	adantr 482	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐶 ∈ Cat)
11	6	adantr 482	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐷 ∈ Cat)
12		diag1f1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
13		diag1f1.0	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ ∅)
14	13	adantr 482	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐵 ≠ ∅)
15		simprl 777	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
16		simprr 779	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
17		eqid 2741	. . . 4 ⊢ ((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑥)
18		eqid 2741	. . . 4 ⊢ ((1^st ‘𝐿)‘𝑦) = ((1^st ‘𝐿)‘𝑦)
19	4, 10, 11, 1, 12, 14, 15, 16, 17, 18	diag1f1lem 49808	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑦) → 𝑥 = 𝑦))
20	19	ralrimivva 3184	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑦) → 𝑥 = 𝑦))
21		dff13 7201	. 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 590	1 ⊢ (𝜑 → (1^st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∀wral 3055 ∅c0 4263 ⟶wf 6484 –1-1→wf1 6485 ‘cfv 6488 (class class class)co 7359 1^st c1st 7931 2^nd c2nd 7932 Basecbs 17174 Catccat 17625 Func cfunc 17816 FuncCat cfuc 17907 Δ_funccdiag 18173

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-om 7810 df-1st 7933 df-2nd 7934 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-fz 13457 df-struct 17112 df-slot 17147 df-ndx 17159 df-base 17175 df-hom 17239 df-cco 17240 df-cat 17629 df-cid 17630 df-func 17820 df-nat 17908 df-fuc 17909 df-xpc 18133 df-1stf 18134 df-curf 18175 df-diag 18177

This theorem is referenced by: eufunclem 50023 diag1f1o 50036

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