diag1f1 - Mathbox for Zhi Wang

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Theorem diag1f1 49629

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 17892	. . 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 18169	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
8	7	func1st2nd 49398	. . 3 ⊢ (𝜑 → (1^st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2^nd ‘𝐿))
9	1, 3, 8	funcf1 17795	. 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 49628	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑦) → 𝑥 = 𝑦))
20	19	ralrimivva 3180	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑦) → 𝑥 = 𝑦))
21		dff13 7203	. 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 4286 ⟶wf 6489 –1-1→wf1 6490 ‘cfv 6493 (class class class)co 7361 1^st c1st 7934 2^nd c2nd 7935 Basecbs 17141 Catccat 17592 Func cfunc 17783 FuncCat cfuc 17874 Δ_funccdiag 18140

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683 ax-cnex 11087 ax-resscn 11088 ax-1cn 11089 ax-icn 11090 ax-addcl 11091 ax-addrcl 11092 ax-mulcl 11093 ax-mulrcl 11094 ax-mulcom 11095 ax-addass 11096 ax-mulass 11097 ax-distr 11098 ax-i2m1 11099 ax-1ne0 11100 ax-1rid 11101 ax-rnegex 11102 ax-rrecex 11103 ax-cnre 11104 ax-pre-lttri 11105 ax-pre-lttrn 11106 ax-pre-ltadd 11107 ax-pre-mulgt0 11108

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 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 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 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-er 8638 df-map 8770 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12407 df-z 12494 df-dec 12613 df-uz 12757 df-fz 13429 df-struct 17079 df-slot 17114 df-ndx 17126 df-base 17142 df-hom 17206 df-cco 17207 df-cat 17596 df-cid 17597 df-func 17787 df-nat 17875 df-fuc 17876 df-xpc 18100 df-1stf 18101 df-curf 18142 df-diag 18144

This theorem is referenced by: eufunclem 49843 diag1f1o 49856

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