Theorem nateq 50549

Description: Two natural transformations with the same source and target are equal exactly when all their components agree. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypotheses

Ref	Expression
isnat3.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
isnat3.b	⊢ 𝐵 = (Base‘𝐶)
natfn2.a	⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
nateq.l	⊢ (𝜑 → 𝐿 ∈ (𝐹𝑁𝐺))

Assertion

Ref	Expression
nateq	⊢ (𝜑 → (𝐴 = 𝐿 ↔ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) = (𝐿‘𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐿   𝜑,𝑥

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)   𝐹(𝑥)   𝐺(𝑥)   𝑁(𝑥)

Proof of Theorem nateq

Step	Hyp	Ref	Expression
1		isnat3.n	. . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷)
2		isnat3.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		natfn2.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
4	1, 2, 3	natfn2 50548	. 2 ⊢ (𝜑 → 𝐴 Fn 𝐵)
5		nateq.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝐹𝑁𝐺))
6	1, 2, 5	natfn2 50548	. 2 ⊢ (𝜑 → 𝐿 Fn 𝐵)
7		eqfnfv 7052	. 2 ⊢ ((𝐴 Fn 𝐵 ∧ 𝐿 Fn 𝐵) → (𝐴 = 𝐿 ↔ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) = (𝐿‘𝑥)))
8	4, 6, 7	syl2anc 596	1 ⊢ (𝜑 → (𝐴 = 𝐿 ↔ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) = (𝐿‘𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∀wral 3080   Fn wfn 6553  ‘cfv 6558  (class class class)co 7443  Basecbs 17416   Nat cnat 18148

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-ixp 8954  df-func 18062  df-nat 18150

This theorem is used by:  natideq  50550  natcoeq  50551