diag1f1 - Mathbox for Zhi Wang

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Theorem diag1f1 49770

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 2735	. . . 4 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
3	2	fucbas 17919	. . 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 18196	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
8	7	func1st2nd 49539	. . 3 ⊢ (𝜑 → (1^st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2^nd ‘𝐿))
9	1, 3, 8	funcf1 17822	. 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 2735	. . . 4 ⊢ ((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑥)
18		eqid 2735	. . . 4 ⊢ ((1^st ‘𝐿)‘𝑦) = ((1^st ‘𝐿)‘𝑦)
19	4, 10, 11, 1, 12, 14, 15, 16, 17, 18	diag1f1lem 49769	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑦) → 𝑥 = 𝑦))
20	19	ralrimivva 3178	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑦) → 𝑥 = 𝑦))
21		dff13 7198	. 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 2930 ∀wral 3049 ∅c0 4263 ⟶wf 6483 –1-1→wf1 6484 ‘cfv 6487 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 Basecbs 17168 Catccat 17619 Func cfunc 17810 FuncCat cfuc 17901 Δ_funccdiag 18167

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 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 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 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8632 df-map 8764 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-fz 13451 df-struct 17106 df-slot 17141 df-ndx 17153 df-base 17169 df-hom 17233 df-cco 17234 df-cat 17623 df-cid 17624 df-func 17814 df-nat 17902 df-fuc 17903 df-xpc 18127 df-1stf 18128 df-curf 18169 df-diag 18171

This theorem is referenced by: eufunclem 49984 diag1f1o 49997

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